To my grandparents

Acknowledgments I am indebted to my doctoral supervisors Valentin Poenaru and Corrado Tanasi. It has been a privilege to have been introduced to Low-Dimensional Topology and Geometric Group Theory. The possibility they give me to work on a “co-tutel” thesis has been a great opportunity to improve my mathematical knowledge and even my personal experience. A special thanks goes to Louis Funar for his guidance. Without him I would have never been able to reach where I am now. He communicates the subject with great insight, enthusiasm and clarity, and has been a constant source of good advice and suggestions of research problems to tackle. I am also grateful to him for reading early drafts of much of the material in this thesis and for his useful comments and corrections. I am also deeply indebted to Pierre Pansu for his disposability and for the time he spent to give me helpful explications during my stay in Paris. I thank Tullio G. Ceccherini-Silberstein and Michel Boileau for having accepted to be my referees. Their remarks have been precious. I thank also Bertrand Remy for carefully reading my work and for many advices. A special acknowledgment goes to all the people at the Laboratoire de Mathematique d’Orsay and at the Institut Fourier of Grenoble, for their support during these past few years. They have contributed a lot in my work. In particular I am grateful to Frederic Haglund, Fran¸cois Labourie, Frederic Paulin, Panos Papasoglou, Estelle Souche, G´erard Besson, Sylvestre Gallot, Bertrand Remy for their stimulating conversations and helpful comments, Martine Justin, Val´erie Lavigne, Marie-Christine Myoupo, Adrien Ramparison, Myriam Charles, Micka¨el Marchand, Fran¸coise Martin and Corinne Sallustio for their assistance, as well as to Vincent Bayle for the football matches we have played at the campus of Grenoble. I want also thank Giusi Castiglione, Giosu´e Lo Bosco, Vincenzo Sciacca and Cesare Valenti for their help within the Dipartimento di Matematica di Palermo. I am also grateful to Goulnara Arzhantseva, Jim Cannon, Cornelia Drut¸u, Tim Riley and Mark Sapir for their interesting comments during a conference in Geneva. A big merci to Maxime Wolff for his innumerable comments. Many progress in writing this thesis was provided by the stimulating discussions we had about my researches. I would also like to thank Giancarlo Passante, who was my advisor at the Universit´a di Palermo, where I started studying mathematics. I owe a great deal to his care and dedication in teaching.

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A final thought goes to all the people who have supported me during these years and with whom I shared many marvelous moments, in particular my brother Fabio, Gabriele Fici, Graham Smith, and all my friends of the Cit´e Universitaire de Paris, as well as Simone Diverio and Fabio Zuddas in Grenoble. Having said all this, there are others to thank, but not in the mathematical sense... Simply because they make my life happy. I am lucky having a family like mine. I cannot express in words how grateful I am. Their enormous support, love and encouragement accompany me always. My father’s advice, my mother’s strength and courage, and my brother’s friendship are a treasure for me. Joanna, just thanks for all...

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0.0.1

Abstract

The general subject of this thesis is the geometric group theory: quasiisometry invariants of finitely presented groups whose definitions are inspired by low-dimensional topology (at infinity). The central notions considered are: the simple connectivity at infinity (sci ), the sci growth, the end-depth, the weak geometric simple connectivity, the Tucker property and their invariance. A group is said to be sci if some (any) presentation complex X is sci, i.e. any e is contained in a compact L such that loops outside L can compact K in X be filled outside K. The sci growth, V1 , is the growth of the smallest function V1 (r) such that loops outside the ball of radius V1 (r) are null-homotopic outside the ball B(r). We first show that the class of V1 is a quasi-isometry invariant for groups; moreover V1 is linear for many cocompact (and some non-cocompact) lattices in connected Lie groups, 3-manifolds groups, Coxeter and Artin groups. The same kind of result is proved for the end-depth (the 0-dimensional analogous of the sci growth), a new end-invariant of oneended groups. We also prove that amalgamated products of sci groups over a subgroup with nontrivial end-depth have nontrivial sci growth. In the second part, we study the relationships between the geometric simple connectivity (no 1-handles in a proper handlebody decomposition), the weak geometric simple connectivity (to be an ascending union of compact simply connected subspaces) and Brick’s quasi simple filtration. We prove that these are equivalent notions for groups, and we provide many examples of wgsc groups. For non-simply connected spaces, we consider the Tucker property (the condition of having only finitely many 1-handles), and we prove its proper homotopy invariance. Key words Quasi-isometries; Asymptotic invariants; Simple connectivity at infinity; (weak) Geometric simple connectivity; Ends; End-depth; Tucker property; Proper homotopy invariants.

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R´ esum´ e Le sujet de cette th`ese est la th´eorie g´eom´etrique des groupes: invariants de quasi-isometries de groupes (de pr´esentation finie) d´efinis `a partir de la topologie (`a l’infini) en petite dimension. Les notions centrales que l’on consid`ere sont: la simple connexit´e ` a l’infini (sci ), la croissance de la sci, la profondeur du bout, la simple connexit´e g´eom´etrique faible (wgsc), la propri´et´e de Tucker et leur invariance. Un groupe est dit sci si le complexe X associ´e `a l’une (toutes) de ses pr´esentations, est sci, c.`a.d. tout compact K de e est contenu dans un compact L tel que tout lacet en dehors de L est trivial X e − K. La croissance de la sci est la croissance de la plus petite fonction en X V1 (r) telle que tout lacet en dehors de la boule de rayon V1 (r) est trivial en dehors de la boule B(r). Nous montrons d’abord que la croissance de la sci est un invariant de quasi-isometries des groupes, et qu’elle est lin´eaire pour la plupart de r´eseaux uniformes (et certains non-uniformes) dans les groupes de Lie, et pour les groupes de 3-vari´et´es, les groupes de Coxeter et d’Artin. Les mˆemes types de r´esultats sont obtenus pour la profondeur du bout (l’´equivalent, en dimension 0, de la croissance de la sci), un nouveau invariant `a l’infini de groupes `a un seul bout. Nous d´emontrons aussi que le produit amalgam´e de groupes sci sur un sous-groupe ayant une profondeur du bout non lin´eaire, a une croissance de la sci non lin´eaire. Dans la deuxi`eme partie, on s’int´eresse aux liens entre la simple connexit´e g´eom´etrique (admettre une d´ecomposition en anses sans anses d’indice 1), la wgsc (ˆetre r´eunion de compacts simplement connexes) et la filtration quasi simple de Brick. On montre qu’il s’agit de propri´et´es ´equivalentes pour les groupes, et on donne plusieurs exemples de groupes wgsc. Pour les espaces non-simplement connexes, on consid`ere la propri´et´e de Tucker (admettre une d´ecomposition avec un nombre fini d’anses d’indice 1), et l’on prouve son invariance par homotopies propres.

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0.1

Introduction

In this thesis we study some asymptotic invariants of discrete groups. The invariants we are looking for encapture part of the behavior at infinity of these universal covering spaces. Thus, they do not say anything interesting for finite groups. This philosophy is due to M. Gromov: given an algebraic problem, translate it into geometry to make it “visible”. The algebraic problem of concern below is to understand finitely-generated groups. The aim is to show how geometric methods may also be important for algebraists. The idea is to associate to any group a metric space on which it acts, and find group properties inherited by such an action. Given a finitely generated group, there exists a natural metric on it, called the word metric. This metric depends on the generating system one chooses. However, the word metric associated to distinct generating sets are similar, in the sense that these are quasi-isometric. We recall that a quasi-isometry between two metric spaces (X, dX ) and (Y, dY ) is a map f : X → Y such that λ−1 dX (x1 , x2 ) − C ≤ dY (f (x1 ), f (x2 )) 6 λdX (x1 , x2 ) + C, ∀y ∈ Y,

and

∃x ∈ X such that dY (y, f (x)) ≤ C,

for some fixed positive constants C and λ. Since it is not possible to classify finitely-presented groups (the word problem is undecidable), Gromov’s aim is to classify them up to quasi-isometries. Hence, interesting properties of groups are those invariants under quasi-isometries (such a property is called geometric or asymptotic). On the other hand, in the 1980’s W.P. Thurston established a firm link between classical geometry and low-dimensional topology. His work made it very clear that the dominating invariant in the study of 3-dimensional manifolds is the fundamental group and that the nature of the fundamental group is very much a function of the geometry associated to the manifold. Beside the Poincar´e conjecture, one of the most important conjectures in lowdimensional topology is the Universal Covering Conjecture: If M is a closed, orientable, irreducible 3-manifold with π1 (M ) infinite, then the universal covering space of M is homeomorphic to R3 . The previous conjecture fails in higher dimension: there exist for any 5

n ≥ 4 closed, aspherical1 manifolds not covered by the Euclidean space ([18]). For this remarkable result, M. Davis uses a fine property of Rn : the simple connectivity at infinity. A space X is simply connected at infinity (abbreviated sci ) if for any compact subset K ⊂ X there exists a compact subset H ⊂ X containing K such that loops outside H can be filled by disks lying in X−K (one may think that loops at infinity bounds disks “near” infinity). Actually, Davis showed that, in any dimension n ≥ 4, there exists a compact, closed, aspherical n-manifold whose universal covering space is not simply connected at infinity. This suffices since a necessary and sufficient condition for an open contractible manifold to be homeomorphic to the Euclidean space is the simple connectivity at infinity. More precisely, in dimension n ≥ 5, J. Stallings and L. Siebenmann ([68, 67]) used the simple connectivity at infinity to characterize the Euclidean space among contractible open topological manifolds (notice that Stallings uses a slightly stronger notion of the sci that allow him to have the same result in all categories TOP, PL and DIFF). In dimension 4 Freedman ([26]) proved that this is true only topologically (there exist exotic structures on R4 ); while in dimension 3 it a classical result ([25]) after assuming the irreducibility to avoid the Poincar´e conjecture. Poenaru in [59] and [60], gave a partial solution to the 3-dimensional covering conjecture; namely he proved that if the fundamental group of a closed 3-manifold is Gromov-hyperbolic, almost-convex or combable, then its universal cover is simply connected at infinity (there are also some improvements by Poenaru-Tanasi in [62, 63]). One of Poenaru’s techniques is to try to “kill” 1-handles at least stably, i.e. after stabilizing the manifold by a product with a n-disk. He shows that the fundamental group at infinity is an obstruction for killing stably the handles of index one; this implies for examples that if W h is the standard 3-dimensional Whitehead manifold (the most important 3-dimensional, contractible, non simply connected at infinity manifold), then W h × Dn can never be represented without 1-handles. Manifolds that admit a handlebody decomposition without 1-handles are called geometrically simply connected (or gsc). In dimension 3 the geometric simple connectivity and the simple connectivity at infinity are closely related since an open gsc 3-manifold is sci. However in higher dimensions there are several open manifolds having a handlebody decomposition without 1-handles that are not simply-connected at infinity. An extension of this concept in the realm of polyhedra (relevant only in 1

A manifold is aspherical if its universal covering is contractible.

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the non-compact case) is the weak geometric simple connectivity. A (possibly non-compact) polyhedron P is weakly geometrically simply connected (abbreviated wgsc) if it admits an exhaustion by compact, connected and simply connected sub-polyhedra. Alternatively, any compact subspace is contained in a simply connected sub-polyhedron. The wgsc is much more flexible than the gsc, the latter making sense only for manifolds, and enables us to work within the setting of polyhedra, and hence for groups (however the wgsc and the gsc are equivalent for non-compact n-manifolds with n 6= 4 [30]). Further developments suggest a generalization of the covering conjecture in higher dimensions as follows: Conjecture 0.1.1. Universal covering spaces of closed, aspherical manifolds are weakly geometrically simply connected. Finally, the right extension of the wgsc to non-simply connected spaces, which is suitable for applications to 3-manifolds, is the Tucker property: a manifold M is Tucker if, for any K compact of M , the fundamental group π1 (M − K) is finitely generated (for each component of M − K). It turns out that this condition can be formulated as a group-theoretical property for coverings. While the geometric simple connectivity is the property of having no 1-handle, the Tucker property expresses the fact that some handlebody decomposition needs only finitely many 1-handles, without any control on their number.

0.1.1

Statements of the results

In this thesis we shall look at the properties defined before, but in the setting of finitely presented groups. Following Brick, Mihalik and Tanasi ([49, 4, 71], we say that a group G is simply connected at infinity (and we write π1∞ G = 0) if the universal covering of some (or equivalently any) compact complex X having G as fundamental group is simply connected at infinity (similar definitions will be used for the wgsc and the Tucker property). We shall prove that being sci or not is a well-defined quasi-isometric notion for groups (our proof is more geometric then that of Brick [4]). Furthermore, we refine this property by defining a function measuring “in which way” a group is sci (see [31]). Definition 0.1.2. Let X be a simply connected non-compact metric space which is simply connected at infinity (i.e. π1∞ X = 0). The rate of vanishing of π1∞ , denoted V1 (r), is the infimal N (r) with the property that any loop 7

which sits outside the ball B(N (r)) of radius N (r) bounds a 2-disk outside the ball B(r). After defining an appropriate equivalence relation on functions, we shall look at the growth of such a function V1 for groups (that we call equivalently the sci growth, the rate of vanishing of π1∞ or just V1 ). Our first result is: Theorem 0.1.3 (2.2.2). The simple connectivity at infinity and the (equivalence class of the) rate of vanishing of π1∞ are quasi-isometry invariants for groups. Furthermore, most of groups coming from geometry are sci in a trivial way (i.e. the function V1 is linear). In particular we have the following: Theorem 0.1.4 (sections 2.3 and 2.4). The function V1 is linear for many cocompact lattices in connected Lie groups; in particular for semi-simple, nilpotent and several classes of solvable Lie groups. This is also true for a special class of non-uniform lattices in higher rank Lie groups (those of Q-rank one). As a corollary we obtain: Corollary 0.1.5 (2.3.9). All geometric 3-manifold groups are simply connected at infinity in a trivial way (i.e. with a trivial V1 ). Moreover, by using the techniques of Davis and Meier [20], one has: Proposition 0.1.6 (section 2.5). Simply connected at infinity Coxeter groups, Artin groups and Euclidean buildings have linear sci growth.

On the other hand one expects that there exists groups with nonlinear V1 ; but the stability of such a function makes the construction of explicit examples to be highly nontrivial. In the aim of finding such groups, we approach the problem from a different perspective: at the lower level. We define an analogous 0-dimensional of the sci growth: the end depth for oneended groups (definition 3.1.1). It turns out that this new invariant has some promising relationships with the sci growth; on the other hand it can be used to distinguish one-ended groups. Ends are one of the most important invariants of finitely generated groups. The celebrated results of Hopf and Stallings ([8] and [70]) have reduced the 8

study of infinitely ended groups to that of one-ended groups. Recall that a space X has one end if for any compact subset C ⊂ X there exists another compact subset D containing C with the property that any two points outside D can be joined by a path in X − C. For a group G, one has to look at the Cayley graph of G with respect to some generating set. Define now the function V0 (r) = inf(R) with the property that any two points which sit outside the ball of radius R centered at the identity e, can be joined by a path outside the ball B(e, r). In the last few years the interest in finding groups with infinite such functions growths (namely for the so-called deadends elements); but all the properties found (as the unbounded dead-end depth [14]) depend on a particular group presentation. In order to provide the right framework to study the connectivity at infinity we are forced to consider some coarse equivalent class of functions. Define then the end depth of G as the growth of the function V0 . Our main result of the third chapter is the following: Theorem 0.1.7 (3.1.3, 3.1.10, 3.1.13, sections 3.1.1 and 3.1.2). 1. The end depth is a well-defined asymptotic invariant for one-ended groups. 2. Hyperbolic groups and CAT (0) groups have a linear end depth. 3. If G1 , G2 are one-ended with linear V0 then the amalgamated free product G = G1 ∗H G2 has one end with a linear end depth. 4. If G1 , G2 are sci with linear V1 and if H is one-ended with linear enddepth, then the amalgamated free product G = G1 ∗H G2 is sci with a linear sci growth.. 5. If G = A ∗H B is the free product with amalgamation over a (sub)group H which is one-ended with a super-linear end depth, then G is simply connected at infinity with a super-linear sci growth. The last statement of the theorem is an interesting link between these two invariants. It should be useful in the following sense: it is not known if there exist groups with a non-trivial sci growth, but it is likely that oneended groups with a super-liner end depth exist (although the example of [14] still furnish a group with linear V0 ).

On the other hand, we shall be interested in a translation of the idea of concealing 1-handles for manifolds in the realm of groups. The concept of 9

the geometric connectivity using handle theory was developed by Poenaru in his work concerning the Poincar´e conjecture and the covering conjecture. The original motivation was that Casson had developed an idea about the metric geometry of the Cayley graph of a group having to do with the covering conjecture in dimension three. Casson’s proof involved “approximating” the universal covering by compact, simply-connected three-manifolds. This condition was then adapted for groups by Brick and Mihalik in [3] (see definition 4.2.17). They called this property quasi-simple filtration (or qsf ). Since it seems very difficult (and plausibly undecidable) to check whether a given non-compact polyhedron (or complex) verifies the qsf condition of Brick, we have considered a related and apparently stronger notion which is the weak geometrical simple connectivity defined before. A group is said wgsc if there exists a compact polyhedron whose universal covering is wgsc (definition 4.2.6). This property depends on the presentation of the group (actually we can show that any finitely generated group with an element of infinite order admits a presentation whose Cayley 2-complex is not wgsc). Nevertheless, we shall prove that these two notions are actually equivalent for discrete groups i.e. for the universal coverings (corollary 4.2.31). This also yields the following characterization of the wgsc: the group Γ is wgsc fn of any compact manifold M n with if and only if the universal covering M π1 (M n ) = Γ and dimension n ≥ 5 is wgsc. The problem of finding aspherical non wgsc manifolds which are universal coverings seems to be much more difficult. It is worthy to note that all reasonable example of groups are wgsc. In fact: Theorem 0.1.8 (section 4.3). 1. Let H be a finite index subgroup of a group G. Then G is wgsc if and only if H is wgsc. 2. Let A and B be wgsc groups and C be a common finitely generated subgroup. The amalgamated free product G = A∗C B is wgsc. Moreover, if φ denotes an automorphism of a wgsc group A, then the HNN-extension G = A ∗C φ is wgsc. 3. All one-relator groups are wgsc. 4. Simply connected at infinity groups are wgsc. 5. Tame combable groups are wgsc. In particular, almost convex groups are wgsc. 10

6. Hyperbolic groups and, more generally, groups acting geometrically on a CAT(0) space are wgsc. 7. Baumslag-Solitar groups are wgsc. 8. Higman group is wgsc. 9. Groups admitting a complete geodesic rewriting system are wgsc. It is not known if there exists finitely presented groups which are not wgsc. Actually, as already noticed, the following statement might well be true (this will be a far reaching generalization of the 3-dimensional covering conjecture): fundamental groups of closed aspherical manifolds are wgsc.

As already noticed, if one extends the notion of wgsc to non-simply connected manifolds, one has to recall the Tucker property (definition 5.1.1), which may be seen as the existence of a handlebody decomposition with finitely many 1-handles. This condition was studied by W. Tucker in dimension three. He proved that this condition is equivalent for a 3-manifold to be tame (i.e. it is the interior of a 3-manifold with boundary). Later, Mihalik and Brick extended this property for groups, by showing also its quasi-isometry invariance (see [6] and [5]). On the other hand, Poenaru, in the 1980’s, invented a useful machinery (which he called the Φ/Ψ theory) in connection with a new viewpoint about the universal covering space of a 3-manifold ([58, 59]. This allowed him to prove, for example, that an open three-manifold is simply connected at infinity whenever the product of it with an n-ball is geometrically simply connected. This criterion is an essential ingredient in Poenaru’s work. The idea is to obtain information in dimension 3 by giving conditions in higher dimensions. To do this, it is necessary to have some proper homotopy invariant. We have then applied these techniques for non simply-connected manifolds (without any restriction on the dimension). In chapter five we shall prove that: Theorem 0.1.9 (5.1.3). The Tucker property is a proper homotopy invariant. As a corollary one obtains a Poenaru-type theorem: a manifold is Tucker whenever the product of it with an n-ball is Tucker. 11

Furthermore, we use the same technique (theorem 5.5.1) to show that Brick’s qsf property is also a proper homotopy invariant (while for the wgsc one has to exclude the dimension 4, as showed in [30]). The organization of the thesis is the following. In chapter 1 we shall introduce the necessary background of the Geometric Group Theory: finitely-presented groups, Cayley graph, ends of groups, quasi-isometries and some typical asymptotic invariants. In chapter 2, we shall define and study the simple connectivity at infinity of groups and its rate of vanishing, by proving as well, its quasi-isometry invariance. Several examples of groups with linear sci growth will be given. In chapter 3, we shall be interested in the end depth of groups. We shall describe some relations between the end depth and the sci growth (in particular for amalgamated free product). The chapter 4 will be devoted to groups that are (weakly) geometrically simply connected. We shall compare this notion with other related properties (qsf and Dehn-exhaustibility) by showing that these are all equivalent in the category of groups. We shall give many examples of wgsc groups. Finally, in chapter 5, we will focus on the Tucker property. After an introduction to Poenaru’s Φ/Ψ theory, the proper homotopy invariance of the Tucker condition will be proved. We have added two appendices to introduce Davis’ example of aspherical manifolds not covered by the Euclidean space (appendix A), and some generalities on lattices in Lie groups (appendix B). At the end, we have inserted a reminder for some of the recurrent definitions, with their mutual relations.

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Contents 0.1

0.0.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1.1 Statements of the results . . . . . . . . . . . . . . . . .

1 Definitions and basic properties 1.1 Cayley graphs and quasi-isometries . . . . . . . 1.2 Finitely presented groups as fundamental groups 1.2.1 Classifying space . . . . . . . . . . . . . 1.2.2 Amalgamated free products . . . . . . . 1.2.3 HNN extensions . . . . . . . . . . . . . . 1.3 Geometric properties . . . . . . . . . . . . . . . 1.3.1 Ends . . . . . . . . . . . . . . . . . . . . 1.3.2 Hyperbolic groups . . . . . . . . . . . . 1.3.3 CAT(0) spaces and the visual boundary 2 Simple connectivity at infinity 2.1 The simple connectivity at infinity for groups 2.1.1 Examples . . . . . . . . . . . . . . . . 2.1.2 Rate of vanishing of π1∞ . . . . . . . . 2.2 Quasi-isometry invariance . . . . . . . . . . . 2.3 Uniform lattices in Lie groups . . . . . . . . . 2.4 Non-uniform lattices . . . . . . . . . . . . . . 2.5 Coxeter groups . . . . . . . . . . . . . . . . . 2.5.1 A proof using the semistability . . . . 2.5.2 A proof using Davis-Meier criterion . . 2.5.3 Buildings . . . . . . . . . . . . . . . . 2.6 The fundamental group at infinity . . . . . . .

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3 The end depth 49 3.1 On distinguishing one-ended groups . . . . . . . . . . . . . . . 49 13

3.1.1 3.1.2

Amalgamated free product . . . . . . . . . . . . . . . . 55 HNN-groups . . . . . . . . . . . . . . . . . . . . . . . . 57

4 The weak geometric simple connectivity 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Definition of wgsc groups . . . . . . . . . . . . . . . . . 4.2.1 The qsf property after Brick and Mihalik . . . . 4.2.2 Small depth and 1-tame groups . . . . . . . . . 4.2.3 Comparison of the various tameness conditions . 4.2.4 The wgsc growth . . . . . . . . . . . . . . . . . 4.3 Some examples of wgsc groups . . . . . . . . . . . . . . 4.3.1 Baumslag-Solitar groups . . . . . . . . . . . . . 4.3.2 Solvable groups . . . . . . . . . . . . . . . . . . 4.3.3 Higman’s group . . . . . . . . . . . . . . . . . . 4.3.4 Further examples . . . . . . . . . . . . . . . . . 4.3.5 From discrete groups to aspherical manifolds . . 4.3.6 Balls and spheres in Cayley graphs . . . . . . . 4.3.7 Rewriting systems . . . . . . . . . . . . . . . . 5 The 5.1 5.2 5.3

Tucker property Introduction . . . . . . . . . . . . . . . . . . . Preliminaries on Poenaru’s Φ/Ψ-theory . . . . The proof . . . . . . . . . . . . . . . . . . . . 5.3.1 Outline of the proof . . . . . . . . . . . 5.3.2 Proof of the theorem using the lemmas 5.4 Proof of the lemmas . . . . . . . . . . . . . . 5.4.1 Proof of the lemma 5.3.4 . . . . . . . . 5.4.2 Proof of the lemma 5.3.6 . . . . . . . . 5.4.3 Proof of the lemma 5.3.7 . . . . . . . . 5.4.4 Proof of the lemma 5.3.8 . . . . . . . . 5.5 Proper homotopy invariance of the qsf . . . .

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A Compact aspherical manifolds not covered by Rn 97 A.0.1 A cubical complex with prescribed links . . . . . . . . 98 A.0.2 The example . . . . . . . . . . . . . . . . . . . . . . . 100 A.0.3 The reflection trick . . . . . . . . . . . . . . . . . . . . 101 B Discrete subgroups of Lie groups 103 B.0.4 Uniform lattices . . . . . . . . . . . . . . . . . . . . . . 103 B.0.5 Non-uniform lattices and Q-rank . . . . . . . . . . . . 104 14

Chapter 1 Definitions and basic properties This chapter is devoted to the introduction of the basic tools of the Geometric Group Theory. We would like to explain how and why it can be useful to do geometry with groups. All groups referred to in this thesis will be infinite, unless the contrary will be stated. The reason is that we will consider universal coverings spaces of compact polyhedra with a given fundamental group. The main references for this chapter are [22], [11], [46], [8], and [35].

1.1

Cayley graphs and quasi-isometries

The fundamental notions that must be understood if one is to be comfortable with geometric group theory are group presentations, Cayley’s graphs and quasi-isometries. A group presentation has the form G =< C|R >. In this presentation the set C is a finite set of letters called the generating set for the presentation. The set C has the property that each letter c ∈ C is paired with another letter c−1 called the inverse letter. We will always assume that C = C −1 . A word in the alphabet C is a finite sequence of elements of C. The set R is a collection of words in the alphabet C and is called the defining relator set for the presentation. The symbol G refers to the group we will define below. Let W denote the set of all words in the generating set C. Let T denote the set of all trivial words, that is words of the form cc−1 , where c ∈ C. Declare two words w and w0 to be equivalents if w0 can be obtained from w by either inserting or deleting at some point a copy of a word from R ∪ T . Extend this notion of equivalence to an equivalence relation ∼. Let G = W/ ∼ denote the set of equivalence classes, and define the product of two such classes a 15

and b to be the class ab represented by the concatenation ab of a and b. It is easy to check that G with this well-defined multiplication is a group, that we will write G =< C|R >. The group G is finitely presented if C and R are finite. (We will always assume that C = C −1 is finite, i.e. G will be always finitely generated ). An easy example of a presentation is represented by the fundamental group of a connected simplicial complex. Indeed, assume that X be such a space, triangulated, with base point x0 a vertex of the triangulation. Collapse a maximal subtree of the 1-skeleton to the base point. Then the image of the 1-skeleton of X is a bouquet of loops, each loop supplying a generator for the fundamental group. Each 2-cell is attached along its boundary to the bouquet by an attaching map which can be realized as a word in the generators and their inverses. These words supply the defining relators of a presentation. If the complex is finite, then the presentation is finite. Finite presentation of groups arise naturally in a wide range of mathematical contexts (e.g. surface groups, 3-manifold groups, Coxeter groups, Artin groups, co-compact lattices in Lie groups, etc.). As already said, we would like to do geometry. Actually, we would like to attach to any group a “good” space, that we will call a geometry. For our aim, a geometry is a topological space endowed with a proper path metric (this is one of the underlying ideas on Gromov approach [38]), namely a metric such that the distance between each pair of points is realized as the length of some path in the space joining these points; the metric is proper if closed metric balls of finite radius are compact. The typical example is a complete Riemannian manifold (by Hopf-Rinow theorem). Let X be a geometry and G a group acting on X. The action is said to be: • isometric if for each g ∈ G and for any x, y ∈ X, d(gx, gy) = d(x, y), • cocompact if the orbit space X/G is compact, • properly discontinuous if the set {g ∈ G : K ∩ gK 6= ∅} is finite for any K compact, • geometric whenever all these three properties hold. Now, let G be a finitely generated group, with neutral element denoted by e. Let S be a finite generating set (for simplicity, we will always assume that e ∈ / S and that S = S −1 ). Define the length lS (g) of any element of G to be the smallest integer n such that there exists a sequence (s1 , s2 , · · · , sn ) of generators in S for which g = s1 s2 . . . sn . Then we can define the distance 16

dS by dS (a, b) = lS (a−1 b). This distance makes G a metric space, and dS is called the word metric with respect to the generating set S. Since dS takes integral values, the space (G, dS ) is discrete, and this may impede geometric understanding. Actually, we want find a “natural” geometry on which a finitely generated group acts. This space is the Cayley graph (see [22] for an extensive discussion). Definition 1.1.1. Let G be a finitely generated group. The Cayley graph C(G, S) of G with respect to a finite generating set S is the graph whose vertices are the elements of G and two vertices g1 , g2 are the two ends of an edge if and only if dS (g1 , g2 ) = 1 (or equivalently g1−1 g2 ∈ S). This graph is infinite whenever G is, and G acts naturally on it by left multiplication. Each edge of C(G, S) can be made a metric space isometric to the segment [0, 1], in such a way the action becomes isometric. One define naturally the length of a path between two points of the graph and the distance between two points is defined as the infimum of the appropriate path-length. In this way we have associated to any infinite group a metric space on which the group acts geometrically. Remark 1.1.2. Actually the Cayley graph is a geometry if and only if the generating set S is finite. Examples: • If G = Z generated by {+1, −1}, then the Cayley graph is isometric to the real line. • The group Z2 generated by {(1, 0), (0, 1), (−1, 0), (0, −1)} has a Cayley graph isometric to the standard square grid in R2 . • The Cayley graph of the free group generated by the set S = ({a, b, a−1 , b−1 } is a 4-valence tree (i.e. with 4 branches coming out from any vertex). The alert reader will note that whenever one changes the generating system, the Cayley graph and the distance dS can change a lot (but, actually, not so deeply...). Indeed, these definitions depend on S. However, if one stands far back, then two Cayley graphs of the same group looks alike, that is: in the large-scale they are the same. This motivates the following definition [38].

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Definition 1.1.3. The metric spaces (X, dX ) and (Y, dY ) are quasi-isometric if there are constants λ > 0,C ≥ 0 and maps f : X → Y and g : Y → X (called (λ, C)-quasi-isometries) so that, for all x, x1 , x2 ∈ X and y, y1 , y2 ∈ Y , the following holds: dY (f (x1 ), f (x2 )) 6 λdX (x1 , x2 ) + C dX (g(y1 ), g(y2 )) 6 λdY (y1 , y2 ) + C dX (gf (x), x) 6 C dY (f g(y), y) 6 C. This relation is an equivalence relation between metric spaces. Observe also that the maps f and g are not necessarily continuous. For example, the real line R and Z are quasi-isometric: it suffices to check with the map f which takes a real number r to its integral part. More generally, any (finitely generated) group G is quasi-isometric to its Cayley graph. Proposition 1.1.4. Let S and S 0 be two finite generating sets of the same group G, and let dS and d0S 0 be the distances defined on G by S and S 0 respectively. Then (G, ds ) and (G, d0S 0 ) are quasi-isometrics. Proof. Let f be the identity map of G. Set λ1 = max {d0S 0 (s, e) : s ∈ S} and λ2 = max {dS (s0 , e) : s0 ∈ S 0 }. It is easy to check that d0S 0 (f (x), f (y)) ≤ λ1 dS (x, y) and similarly, dS (f −1 (x), f −1 (y)) ≤ λ2 d0S 0 (x, y). This ends the proof. This proposition means that, on a finitely generated group, the word metric is unique up to quasi-isometry. It also follows that the Cayley graph associated to a (finitely-generated) group is a well-defined geometry up to quasi-isometry (hence, any quasi-isometry property of the Cayley graph can be viewed as a property of the group itself). More generally, it is an exercise to prove that if a group G acts geometrically on two geometries X and Y , then X and Y are quasi-isometric (see [11]). Remark 1.1.5. • There exist different groups with isomorphic Cayley graphs. For examples, if An , Bn are groups of order n, then the Cayley graphs of the free product Ap ∗ Bk and Bp ∗ Ak are the same, though these groups are, in general, not isomorphic. 18

• A metric space if quasi-isometric to a point if and only if its diameter is finite. In particular, finite groups are all quasi-isometric to the trivial group. The next theorem is the fundamental observation in geometric group theory. We sketch a proof, for details see [22]. Proposition 1.1.6. Let X be a geometry and G be a group acting geometrically (i.e. cocompactly, isometrically and proper discontinuously). Then G is finitely generated and quasi-isometric to X. Proof. Let π : X → X/G be the canonical projection. The space X/G has a canonical metric defined by d(p, q) = inf{d(x, y) : x ∈ π −1 (p) and y ∈ π −1 (q)}. As X/G is compact, its diameter R is finite. Choose a base point x0 in X and set B = {x ∈ X : d(x0 , x) ≤ R}. Set S = {g ∈ G : g 6= e, and gB ∩ B 6= ∅}. Observe that S = S −1 and that it is finite, since the action is proper discontinuous. Finally, set r = inf{d(B, gB) : g ∈ G − (S ∪ e)}. One can prove that S generates G and that dS (e, g) = r−1 d(x0 , gx0 ) + 1. Consider now the map f : G → X sending an element g to the point gx0 . We have that dS (g1 , g2 ) ≤ r−1 d(f (g1 ), f (g2 )) + 1. Furthermore, since d(x0 , gx0 ) ≤ λdS (e, g) for all g ∈ G, where λ = sup{d(x0 , sx0 ) : s ∈ S}, one has that dS (g1 , g2 ) ≤ λdS (g1 , g2 ). To finish, we have that d(f (G), x) ≤ R for all x ∈ X because (gB)g∈G covers X. Corollary 1.1.7. Let H be a finite index subgroup of a finitely generated group G. Then H is finitely generated and quasi-isometric to G. Another example of a space constructed in relation to a group is the Rips complex. Let X be a metric space and let r be a positive number. The Rips complex of X of parameter r is the simplicial polyhedron Ripsr (X) with set of vertices X itself, and, for each dimension k ≥ 1, with k-simplexes the subsets {x0 , x1 , . . . , xk } of X whose diameter is bounded by r. The space Ripsr (X) is a topological space (for the weak topology). Whenever X is a group G generated by S, with the word metric dS , the Cayley graph is precisely the 1-skeleton of Rips1 (G, S). It is not difficult to prove that a finitely generated group is finitely presented if and only if Ripsr (G, S) is simply connected for r large enough [35].

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1.2

Finitely presented groups as fundamental groups

Let G be a finitely presented group. We want to construct a manifold whose fundamental group is G. To do this, we begin with the Cayley 2-complex associated to a group presentation. Let P =< s1 , s2 , . . . , sn |r1 , rs , . . . , rt > be a presentation for G. Define a finite 2-complex XG as follows: • the 0-skeleton is a point • the 1-skeleton is a wedge of n oriented circles • the complex XG is obtained from the 1-skeleton by attaching one 2-cell for any relation ri . The Van Kampen theorem shows that G is exactly the fundamental group of this complex. The universal covering C(P) of XG is the Cayley 2-complex of G with respect to the presentation P. Observe that the 1-skeleton of C(P) is the Cayley graph of G generated by (s1 , s2 , . . . , sn ). One can easily show that a 2-complex XG associated to a group presentation embeds in R5 . Let M be the boundary of a small neighborhood N of the image j(XG ) of XG . The fundamental groups of XG and N are isomorphic since N retracts by deformation onto j(XG ). By transversality, the complement N − j(XG ) has the same π1 as N , since j(XG ) has codimension 3. Finally, the complement N − j(XG ) deformation retracts onto M . Then M is a smooth 4-manifold without boundary and whose fundamental group is isomorphic to G.

1.2.1

Classifying space

In this section we introduce a class of spaces whose homotopy type depends only on their fundamental group. These spaces arise many places in topology, especially in its interactions with group theory. A path-connected space whose fundamental group is isomorphic to a given group G and which has a contractible universal covering space is called a K(G; 1)-space. The 1 here refers to π1 . All these spaces are called Eilenberg-MacLane spaces, or classifying space of G. It is not hard to construct a K(G, 1) for an arbitrary group G. Let EG be the complex whose n-simplices are the ordered (n + 1)-tuples [g0 , g1 , . . . gn ] of elements of G. Such an n-simplex attaches to the n − 1simplices [g0 , g1 , . . . gˆi . . . gn ] in the obvious way, just as a standard simplex 20

attaches to its faces. (The notation gˆi means that this vertex is deleted). The complex EG is contractible by the homotopy ht that slides each point x ∈ [g0 , g1 , . . . gn ] along the line segment in [e, g0 , g1 , . . . gn ] from x to the vertex [e], where e is the identity element of G. This is well-defined in EG since when we restrict to a face [g0 , g1 , . . . gˆi . . . gn ] we have the linear deformation to [e] in [e, g0 , g1 , . . . gˆi . . . gn ]. Note that ht carries [e] around the loop [e, e], so ht is not actually a deformation retraction of EG onto [e]. This construction of a K(G; 1) produces a rather large space, since BG is always infinite-dimensional, and if G is infinite, BG has an infinite number of cells in each positive dimension. A different construction of a K(G; 1) is: start with any 2-dimensional complex having fundamental group G, for example the complex XG associated to a presentation of G, and then one attaches cells of dimension 3 and higher to make the universal cover contractible without affecting the π1 . In general, it is hard to get any control on the number of higher-dimensional cells needed in this construction, so it too can be rather inefficient. Indeed, finding an efficient K(G; 1) for a given group G is often a difficult problem. In spite of the great latitude possible in the construction of K(G; 1)’s, there is a very nice homotopical uniqueness property that accounts for much of the interest in K(G; 1) (for a proof see a basic book in Algebraic Topology): Theorem 1.2.1. The homotopy type of a CW complex K(G; 1) is uniquely determined by G. Having a unique homotopy type of K(G; 1)’s associated to each group G means that algebraic invariants of spaces that depend only on homotopy type, such as homology and cohomology groups, become invariants of groups. This has proved to be a quite fruitful idea, and has been much studied both from the algebraic and topological viewpoints. Here is a simple and well-known theorem. Theorem 1.2.2. A group G acts geometrically on a n-connected geometry if and only if there exists a K(G; 1)-space with finite (n + 1)-skeleton.

1.2.2

Amalgamated free products

Consider two groups G1 and G2 presented as follows: G1 is the group with presentation < w1 , . . . , wk , a1 , . . . , an | Q1 , . . . , Qt , R1 , . . . , Rj >, and G2 is generated by {m1 , . . . , md , a1 , . . . , an } with relations {P1 , . . . , Ps , R1 , . . . , Rj }. They have a common subgroup H with generators {a1 , a2 , . . . , an } and relations {R1 , R2 , . . . , Rj }. 21

Definition 1.2.3. The amalgamated free product G = G1 ∗H G2 of G1 and G2 over H is the group with generators {w1 , . . . , wk , m1 , . . . , md , a1 , . . . , an } and relators {Q1 , . . . , Qt , R1 , . . . , Rj , P1 , . . . , Ps }. Suppose H has generators {a1 , a2 , . . . , an } with relations {R1 , R2 , . . . , Rj }. Let X1 , YH and X2 be the standard 2-complexes associated to the presentations of G1 , H and G2 respectively. The space X obtained by attaching X1 e is and X2 along YH has G as fundamental group. Its universal covering X constructed from coset copies of the universal coverings of X1 and X2 which are attached at coset copies of YeH .

1.2.3

HNN extensions

Let G1 be a finitely presented group and let H be a subgroup. Let f : H → G1 be a monomorphism from H into G1 , and let K be the image f (H). Suppose that H is generated by {a1 , . . . , an } and denote by ci the generators f (ai ) of K. Let < b1 , . . . , bm , a1 , . . . , an , c1 , . . . , cn | p1 = 1, . . . , pk = 1 > be a presentation for the group G1 . Definition 1.2.4. The HNN-extension of G1 by f , denoted by G = G1 ∗H , is the group with generators {b1 , . . . , bm , a1 , . . . , an , c1 , . . . , cn , t}, and relations p1 = 1, . . . , pk = 1, c1 = t−1 a1 t, . . . , cn = t−1 an t. Consider the standard 2-complex G1 associated to G1 . It contains two subcomplexes H, K associated to H and K. Consider the space G obtained from a copy of G1 and a copy of H × [0, 1] where H × 0 is identified with the copy of H in G1 and H × 1 is identified with the copy of K in G1 . The e of G can be constructed from coset copies of G e1 universal covering space G e × [0, 1], the universal covering spaces of G1 and H. and H

1.3

Geometric properties

A property (P ) of finitely generated groups is said to be a geometric property if, whenever G1 and G2 are two quasi-isometric (finitely generated) groups, G1 has property (P ) if and only if G2 has property (P ). It is remarkable that there is an abundance of geometric properties, so that quasi-isometry is a very interesting relation between groups: though it ignores finite details, it preserves a lot of distinct properties. We first sketch the proof of the following proposition [22].

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Proposition 1.3.1. Being of finite presentation is a geometric property. Proof. Let G1 and G2 be two quasi-isometric finitely generated groups, with generating sets A1 and A2 . Suppose that the group G2 has a finite presentation < A2 |R2 >. Let Ci be the Cayley graphs of Gi , for i = 1, 2. Put l the length of the longest word in R2 . Thus, the complex K2 obtained by attaching 2-cells to C2 along all edge-loops of length ≤ l is simply-connected. Let f : G1 → G2 and f 0 : G2 → G1 be a (λ, C)- quasi-isometry. Let m = max{λ, C, l} and let M = 3(3m2 + 5m + 1). Construct then a 2complex K1 by attaching 2-cells to the Cayley graph C1 along each edge-loop of length ≤ M . We have to show that π1 K1 = 0. Let now γ be an edge-loop in K1 ; we have to fill it in K1 . The idea is to send it by the quasi-isometry in a loop of K2 . This loop will bound a simplicial disk D. After a triangulation, one can use again the quasi-isometry (on the other sense) to send the vertices of a (sufficiently fine) triangulation of D in K1 . Thanks to the properties of the quasi-isometry, one can show that the image can be filled by 2-cells yielding a disk bounding the original loop. This is due to the fact that close points are sent to close points by a quasi-isometry. One of the oldest quasi-isometry invariant of group is the growth of the group (see [22] for an extensive discussion on this subject). To any group there corresponds a growth function, but to obtain an invariant one needs an appropriate equivalence relation for such functions. Let G be a group with a finite generating set S. The growth function of (G, S) is the function that associates to any k ≥ 0 the number bk of elements of G belonging to the ball B(e, k). One says that two growth functions f1 , f2 are equivalents if there exists two constants c1 , c2 both greater then 1 such that for any t ∈ R one has: f1 (t) ≤ f2 (c1 t) and f2 (t) ≤ f1 (c2 t). For a finitely generated (infinite) group the growth type (i.e. the equivalence class of its growth functions) is well-defined. Two finitely generated groups that are quasi-isometric have the same growth type ([22]). As an application of this result one obtains that Zn and Zm are not quasi-isometric for n 6= m, because the growth type of Zn is tn . Definition 1.3.2. A group G has virtually a property (P ) if it contains a finite index subgroup which has the property (P ).

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Other examples of geometric properties of groups are: being virtually free, virtually cyclic and virtually nilpotent (this is one of the deepest theorem by Gromov in Geometric Group Theory), and the number of ends (see below for a definition). Among properties that are not geometric, one has: being virtually solvable and virtually torsion-free ([24]).

1.3.1

Ends

Let X denote a locally compact, connected metric space. If K is a compact subset of X, then we say that a connected component C of the complementary of K in X is unbounded if its closure in X is noncompact. Denote by e(X, K) the number of unbounded components of X − K. Then one can define the number of ends e(X) of X to be the supremum over all K of the numbers e(X, K) ([11] or [35]). Examples: • a compact metric space has 0 ends; • the real line has 2 ends; • all Euclidean spaces of dimension ≥ 2 are one-ended; • a tree of valence n ≥ 3 has infinitely many ends. Let G be a finitely generated group. The number of ends of G is by definition the number of ends of its Cayley graph (it is easy to check that this number does not depend on the generating set). It takes some more time to show Hopf’s result ([8]) on the number of ends of a group, namely: Theorem 1.3.3 (Hopf ). A group has either 0, 1, 2 or infinitely many ends. The groups with 0 ends are finite. On the other hand, there is a deep result of J. Stallings ([70]) on the structure of group with more then one end, which states: Theorem 1.3.4 (Stallings). A finitely generated group G has exactly two ends if and only if G has an infinite cyclic finite index subgroup. A finitely generated group G has infinitely many ends if and only if G can be factored in one of the two following ways: • G is a free product with finite amalgamating subgroup where this amalgamating subgroup is properly contained in both factors and of index > 2 in at least one factor; 24

• G is a HNN extension amalgamated over a finite subgroup which is properly embedded in base group. This theorem reduced the study of infinitely ended groups to that of oneended groups. Hence, in order to “classify” groups (up to quasi-isometry), one has to look for invariants of one-ended groups in order to better understand the collection of one-ended groups.

1.3.2

Hyperbolic groups

This remarkable class of groups was introduced and developed by Gromov in [37] (see also [35])following his idea to think at infinite groups as geometric objects. Gromov isolated a whole family of equivalent properties of a metric space that are satisfied by manifolds having negative sectional curvature. Then he defined that class of hyperbolic groups which has to be though as a generalization of the geometric properties of groups acting geometrically on hyperbolic non-Euclidean geometry. The most intuitive definition of a hyperbolic metric space is the thin triangles property. A proper path-metric space X has uniformly thin triangles (δ) if, whenever xyz is a geodesic triangle in X and whenever p is a point of one of the three sides, say p ∈ xy, then there is a point q in one of the other two sides such that d(p, q) ≤ δ. Then, a word-hyperbolic group is a group whose Cayley graph has uniformly thin triangles (with respect to some generating system). Observe that fundamental groups of negatively curved spaces are word-hyperbolic. One of the first important (and not obvious) result about hyperbolic group is: Theorem 1.3.5 (Gromov [37]). The hyperbolicity is a quasi-isometry invariant of geodesic metric spaces. Remark 1.3.6. If G is hyperbolic, then one can show that the Rips complex defined above is contractible, and an analysis of the canonical action of G on it shows that G has to be finitely presented (see [35]).

1.3.3

CAT(0) spaces and the visual boundary

The notion of nonpositive curvature makes sense for a more general class of metric spaces than Riemannian manifolds (more on this subject in [8]). A geodesic in a metric space X is a path g : [a, b] → X which is an isometric embedding. The space X is called a geodesic space if any two points can 25

be connected by a geodesic segment. A triangle in a geodesic space X is the image of three geodesic segments meeting at their endpoints. Given a triangle T in X, there is a triangle T ∗ in R2 with the same edge lengths. This triangle T ∗ is called a comparison triangle for T . To each point x ∈ T there is a corresponding point x∗ ∈ T ∗ . The triangle T is said to satisfy the CAT(0)inequality, if given any two points x, y ∈ T we have d(x, y) ≤ d(x∗ , y ∗ ). The space X is nonpositively curved if the CAT(0)- inequality holds for all sufficiently small triangles. Definition 1.3.7. A space X is a CAT(0)-space (or a Hadamard space) if it is complete and if the CAT(0)-inequality holds for all triangles in X. Remark 1.3.8. It follows immediately from the definitions that there is a unique geodesic between any two points in a CAT(0)-space and from this that any CAT(0)-space is contractible. Gromov observed that the universal cover of a complete nonpositively curved geodesic space is CAT(0). Hence, any such nonpositively curved space is aspherical. Finally, we call a group G a CAT(0)-group if it acts geometrically on a CAT(0) space. In contrast with respect to hyperbolic groups, showing that a group is CAT(0) first requires the construction of an appropriate CAT(0) space, and there is no candidate to begin with. Remark 1.3.9. Notice that the definition of hyperbolicity only implies that the large scale “curvature” is negative: we get no information about local structure. On the other hand, CAT(0) spaces have good local “curvature” properties. An useful object to study the geometry at infinity of a CAT(0) space is the boundary at infinity. There are two construction of the boundary ∂X of a CAT(0) space X, the visual description and that of Gromov (arising from a natural embedding of X into the space of continuous functions on X). It turns out that these construction are equivalent. Notice that, in general, ∂X is not a sphere, and, if X is not locally compact, then X ∪ ∂X is not compact (see [8]). Definition 1.3.10. Two geodesic rays r, s : [0, ∞) → X of a metric space X are said to be asymptotic if there exists a constant K such that d(r(t), s(t)) ≤ K for all t ≥ 0. The set ∂X of boundary points of X (or its visual boundary) is the set of equivalence classes of geodesic rays: two geodesic rays are equivalent if and only if they are asymptotic. 26

Remark 1.3.11. Another way to define the visual boundary of X, is to consider ∂X as the set of all geodesic rays emanating from a fixed base point x0 endowed with the compact-open topology (independent of the choice of the base point).

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Chapter 2 Simple connectivity at infinity In this chapter we define the simple connectivity at infinity of groups, and we prove that it is a geometric property of finitely presented groups. We shall also refine this notion introducing a function measuring, roughly speaking, in which way a group is sci. We begin with the following: Definition 2.0.12. A connected, simply connected, locally compact topological space X is simply connected at infinity (and one writes sci or π1∞ X = 0) if for each compact subset k ⊆ X there exists a larger compact subset K, with k ⊆ K ⊆ X, such that any closed loop in X − K is null homotopic in X − k (otherwise we shall write π1∞ X 6= 0). Obviously, the Euclidean space R2 is not simply connected at infinity, while in dimension three the most familiar example of a contractible manifold which is not simply connected at infinity is the Whitehead 3-manifold W h3 (see [73]). Briefly, let V be an unknotted solid torus in S 3 , let hi : V → V be an embedding of V in V such that h(V ) is knotted in V but the meridian of V is null-homotopic in S 3 − h(V ), and let X = the intersection of all hi (V ). Then W h3 = S 3 − X is contractible and non sci; moreover W h3 × R is homeomorphic to R4 .

2.1

The simple connectivity at infinity for groups

We now turn to the definition of the simple connectivity at infinity of groups. Let P = (x1 , x2 , ....., xn |R1 , R2 , ....., Rt ) be a presentation of a group G, where x1 , x2 , ....., xn are the generators of G and R1 , R2 , ....., Rt are the relators of G. Let C(P) be the Cayley 2-complex associated to the presentation P. 28

Definition 2.1.1. A finitely presented group G is said to be simply connected at infinity (or sci) if the Cayley 2-complex C(P), associated to some presentation P of G, is simply connected at infinity. We will show that being sci only depends on the group and not on the presentation. The first observation is that it only depends on the 2-skeleton. Proposition 2.1.2. Let X be a compact, connected, polyhedron and X (2) its e = 0 if and only if π1∞ X e (2) = 0. 2-skeleton. Then π1∞ X e (2) ⊆ X. e Suppose that X e is sci; Proof. Let k be a compact subset of X e verifying definition 2.0.12. then there exists K ⊇ k a compact subset of X The result follows by taking the 2-skeleton K2 of K. Let γ be a loop in e (2) − K2 , then γ is contained in X e − K, and hence it bounds a disk D X e (2) − k. Hence satisfying D ∩ k = ∅. Up to homotopy, D is contained in X e (2) is sci. X e (2) sci, and let c be a compact subset of X. e The Conversely, suppose X (2) e 2-skeleton c2 of c is a compact subset of X ; then there exists a compact e (2) − C2 are null-homotopic in X e (2) − c2 . subset C2 ⊇ c2 such that loops in X e The set C = C2 ∪ {n-cells of c, n ≥ 3} is a compact subset of X containing e − C, then it is homotopically equivalent to a loop in c. If γ is a loop in X e (2) − C2 , by an homotopy of X e − C. Such a loop bounds a disk out of c2 , X (2) e is sci. Hence λ bounds a disk outside c. since X Lemma 2.1.3. If X and Y are two compact, connected 2-dimensional polyhedra with isomorphic π1 ’s, then there exists a compact polyhedron M and compact subpolyhedra X1 and Y1 , such that M collapses onto each of X1 and Y1 ; furthermore, X1 is the wedge of X and a finite number of S 2 ’s, and similarly Y1 is the wedge of Y and a finite number of S 2 ’s. This result goes back to J.H.C. Whitehead ([73]). His proof involved looking at certain moves changing one group presentation into another presentation of the same group: the Tietze transformations (Ti ). • T1 : add r, a consequence of the relators, to the relators; • T2 : the inverse of T1 ; • T3 : add a new generator y and a new relator yu−1 where u is a word in the old generators; • T4 : the inverse of T3 . 29

Explicitly, the collapsing referred to here involves simplicial structures. One says that A collapses to B, when there is a triangulation of A with B covered by sub complexes, and there is a sequence of elementary collapses leading from A to B. An elementary collapse from A to B involves some simplex σ of A not in B, and a face τ of σ which is a face of no other simplex of A (namely a proper face); one then removes the interior of σ and τ to get B. The inverse operation is called an elementary dilatation. Proposition 2.1.4. If X and Y are two compact connected polyhedra with e is sci if and only if Ye is also sci isomorphic π1 ’s, then X Proof. By the previous proposition, we can restrict our to 2-dimensional polyhedra, and, from the previous lemma, it follows that we need to consider only two cases: • Y is the wedge product of X with a 2-sphere, or • Y collapses to X. e and an infinite number of S 2 ’s. In the first case Ye is the wedge product of X e is sci then Ye is also sci (since any loop in Ye based in X e is Obviously, if X e e homotopy equivalent to a loop in X). On the other hand, suppose Y sci and e Then there exists a compact subset KY of let k be a compact subset of X. Ye such that any loop outside KY bounds a disk outside k. If we consider e we obtain a compact subset KX of X e which contains k. Take a KY ∩ X, e e loop γ ⊂ X − KX . Then γ, as a loop of Y − KY , bounds a disk D2 ⊂ Ye − k. e (after removal of some Such a disk D2 can be supposed to be contained in X 2 S ’s), and hence the claim is proved. The second case can be reduced, by induction, to one elementary collapsing. If Y collapses to X by an elementary collapsing at the free face τ e with an infinite numbers of ∆’s and τ ’s. These and simplex ∆, then Ye = X simplexes ∆i are properly embedded in Ye , i.e. they are two by two disjoint and every compact subset intersects only a finite number of them. Let kY be e Suppose that π1∞ X e = 0, then there a compact subset of Ye and kX = kY ∩ X. exists a compact subset KX such that any loop outside KX is null-homotopic e − kX . Put KY = KX ∪ A where A is the set of all ∆i having non-empty in X intersection with KX . Observe that KY is compact since A contains a finite number of elements. Now, let γ be a loop outside this compact subset KY . This loop can be pushed along the interiors of some ∆i (those touched by e − KX . Hence it is null-homotopic in X e − kX and then in λ) to a loop in X Ye − k. This proves the first direction. 30

e Consider cY the On the other hand, let c be a compact subset of X. union of c with all the ∆’s touched by it. It is a compact subset of Ye . By hypothesis there exists another compact subset CY in Ye such that any loop e It is a compact outside CY bounds a disk outside cY . Put CX = CY ∩ X. e Now, any loop outside CX bounds a disk in Ye − cY . This subset of X. means that such a disk does not intersect those ∆’s touched by c, so that, by pushing along some ∆’s, we obtain a disk outside c, as we wanted. e= It follows that if G = π1 X for some compact polyhedron such that π1∞ X e is also sci. 0, then for any other compact polyhedron B with π1 B = G, B This means that π1∞ G = 0 is a well defined group notion. Remark 2.1.5. One can also show directly that Tietze transformations do not affect the simple connectivity at infinity (see for example [71]). Remark 2.1.6. This definition cannot be extended to finitely generated groups with infinitely many relations. In fact, let G be a finitely presented group. For any loop bounding a disk which has to come “near” the origin, one can add a 2-cell (i.e. a relation) permitting to find a null-homotopy far from the origin. Doing this for any such a loop, one obtains a (infinite-dimensional) complex with G as fundamental group, which is simply connected at infinity. Now we want study this class of groups. We start with an easy result. Corollary 2.1.7. If H is a finite index subgroup of G, then π1∞ H = 0 if and only if π1∞ G = 0. Proof. Let X be a compact polyhedron such that π1 X = G with universal e The group G = π1 X acts on X e and so, by restriction, H does covering X. e also. The space X1 = X/H is compact (because the index of H is finite), and the commutative diagram: e X . & e e X/G = X ← X1 = X/H e Hence, by definishows that X and X1 have the same universal covering X. e = 0 ⇔ π1∞ G = 0. tion, π1∞ H = 0 ⇔ π1∞ X

2.1.1

Examples

• As already mentioned, the Euclidean space Rn is simply connected at infinity for n ≥ 3. Hence, if G is an abelian group, then there exists a finite index subgroup H < G such that H = Z + Z... + Z and so π1∞ G = π1∞ H = π1∞ (Z + Z + ...Z) = π1∞ Rn = 0 (for n > 2). 31

• The free group of rank n, Fn = Z ∗ Z ∗ ... ∗ Z is sci. In fact, the space Y = the n-connected sum of (S 1 × S 2 ) has π1 Y = Fn and π1∞ Ye = 0, because Ye = Rn − {tame Cantor set}. (A Cantor set of a manifold M is said tame if it can be embedded into a smooth arc of M ). Observe also that all free groups Fn are quasi-isometric (for n > 1). • A group G quasi-isometric to Zn contains a finite index subgroup isomorphic to Zn (see [35]), and so π1∞ G = π1∞ Zn = 0. • All finite groups are sci (because the Cayley complex of a finite group is compact). • The group Z × (Z ∗ Z) is one-ended but not sci.

2.1.2

Rate of vanishing of π1∞

Now, we want refine the notion of simple connectivity at infinity. We would like to understand in which “fashion” a group becomes sci. The idea is to measure, given a compact subset K which gets larger and larger, how far away one needs to go to kill loops outside K. Definition 2.1.8. Let X be a simply connected non-compact metric space with π1∞ X = 0. The rate of vanishing of π1∞ , denoted V1 (r), is the infimal N (r) with the property that any loop which sits outside the ball B(N (r)) of radius N (r) bounds a 2-disk outside B(r). Remark 2.1.9. It is easy to see that V1 can be an arbitrary large function. It is customary to introduce the following equivalence relation on functions: f ∼ g if there exists constants ci , Cj (with c1 , c2 > 0) such that c1 f (c2 R) + c3 ≤ g(R) ≤ C1 f (C2 R) + C3 . We want show that the equivalence class of V1 (r) is a quasi-isometry invarifG ) will be a quasi-isometry invariant of the ant. In particular V1 (G) = V1 (X fG is the universal covering space of a compact simplicial group G, where X complex XG , with π1 (XG ) = G and π1∞ (G) = 0. Remark 2.1.10. For most groups G coming from geometry V1 (G) is trivial, i.e. linear. Obviously if M has an Euclidean structure then V1 (π1 (M )) is linear. Since metric balls in the hyperbolic space are diffeomorphic to standard balls in Rn one derives that V1 (π1 (M )) is linear for any compact hyperbolic manifold M . 32

Remark 2.1.11. Notice that there exists (see [18]) word hyperbolic groups G (necessary of dimension n ≥ 4 by [3]) which are not simply connected at infinity and hence V1 is not defined. Moreover if G is a word hyperbolic torsion-free group with π1∞ (G) = 0 then it seems that V1 (G) is linear. Remark 2.1.12. The existence of groups G acting freely and cocompactly on Rn , which have super-linear V1 seems most likely. The examples described in ([38], section 4), which have large acyclicity radius, strongly support this claim. The first point is that the rate of vanishing of π1∞ is rather related to higher (i.e. dimension n − 2) connectivity radii, which are less understood. The second difficulty is that these groups are not sci. The simplest way to overcome it is to consider group extensions. For instance π1∞ (G × Z2 ) = 0, for any finitely presented group G; alternatively π1∞ (V n × R) = 0 for any contractible manifold V n (n ≥ 2). However this idea does not work because one can show that V1 (G × Z2 ) is linear.

2.2

Quasi-isometry invariance

For positive d, set Pd (G) for the Rips complex constructed as follows: • its vertices are the elements of G, • the elements x1 ......xn of G span an n-simplex, if d(xi , xj ) ≤ d for all i, j (where d(., .) is the word metric). If G is δ-hyperbolic then Pd (G) is contractible as soon as d > 4δ + 1 (see [35]). But, although Pd (G) is not contractible in general, we can prove that it is simply connected under a mild restriction. Let G =< x1 , ..., xn |R1 , ..., Rp > be a presentation of G and r denotes the maximum length among the relators Ri . Lemma 2.2.1. If 2d > r, then π1 (Pd (G)) = 0. Proof. Let l = [1, γ1 , γ2 , ..., γn , 1] be a (simplicial) loop in Pd (G) based at the identity. Two successive vertices of l are at distance at most d. One can interpolate between two consecutive γj ’s a sequence of elements of G (of length at most d), consecutive ones being adjacent when viewed as elements of the Cayley graph (hence at distance one). The product of elements corresponding to these edges of length Q one of l is trivial in G. Therefore it is a product of conjugates of relators: gi−1 Ri gi . The diameter of each Ri is at most r/2, and the assumption 2d > r, implies that each loop gi−1 Ri gi is contractible in Pd (G). This ends the proof. 33

The natural group action of G on itself by left translations gives rise to an action on Pd (G). In particular, if G has no torsion then it acts freely on Pd (G) and Pd (G)/G = X is a compact simplicial complex with π1 (Pd (G)/G) = G. Our first result is the following: Theorem 2.2.2. The rate of vanishing of π1∞ is a quasi-isometry invariant for finitely-presented groups. We begin to prove a simpler statement, namely: Proposition 2.2.3. The simple connectivity at infinity is a geometric property of torsion-free groups. Proof. One has to show that if the group H is quasi-isometric to the group G then π1∞ Pd (G) = 0 implies that π1∞ Pa (H) = 0 for large enough a. Let f : H → G and g : G → H be (k, C)-quasi-isometries between G and H. Fix x0 ∈ Pa (H) and f (x0 ) ∈ Pd (G) as base points. Lemma 2.2.4. If π1∞ Pd (G) = 0 then π1∞ PD (G) = 0 for D ≥ d. Proof. An edge in PD (G) corresponds to a path (of length uniformly bounded by Dd + 1) in Pd (G). Thus a loop l in PD (G) at distance at least R from a given point corresponds to a loop l0 in Pd (G) at distance at least R − Dd − 1 from the same point. By assumption l0 will bound a 2-disk D2 far away in Pd (G). Now the union of an edge [x1 xn ] in PD (G) and its corresponding path [x1 , . . . , xn ] in Pd (G) ⊂ PD (G) form the boundary of a 2-disk in PD (G), which is triangulated by using the triangles [x1 , xj , xn ]. Consider one such triangulated 2-disk for each edge of l and glue to the previously obtained D2 to get a 2-disk in PD (G) bounding l and far away. By hypothesis for each r there exists N (r) > 0 such that every loop l in Pd (G) satisfying d(l, f (x0 )) > N (r) bounds a disk outside B(f (x0 ), r). This means that there exists a simplicial map ϕ : D2 → Pd (G) − B(f (x0 ), r) such that ϕ(∂D2 ) is the given loop l, when D2 is suitably triangulated. A loop l = [x1 , x2 , ..., xn , x1 ] in Pd (G), based at x1 , is the one-dimensional simplicial sub-complex with vertices xj and edges [xi xi+1 ], i = 1, n (with the convention n + 1 = 1). Set M (R) = kN (kR + kC + 3C) + 3C. We claim that: Lemma 2.2.5. Any loop l in Pa (H) sitting outside the ball B(x0 , M (R)) bounds a 2-disk not intersecting B(x0 , R). 34

Proof. Set l = [x1 , ..., xn ]. Using the previous lemma one can assume that d is large enough such that d−C > 1. As in lemma 2.2.1 one can add extra k vertices between the consecutive ones such that d(xi , xi+1 ) ≤ ε holds, where kε + C = d. The image f (l) = [f (x1 ), ...., f (xn ), f (x1 )] of the loop l has the property that d(f (xi ), f (xi+1 )) ≤ kε+C. Since f is a quasi-isometry, d(x, gf (x)) ≤ C, and then d(gf (x), gf (y)) ≥ d(x, y)−2C, which implies d(x, y) ≤ kd(f (x), f (y))+ 3C and thus: d(f (x), f (y)) ≥

d(x, y) − 3C , for all x, y ∈ Pa (H). k

From this inequality one derives that: d(f (xi ), f (x0 ) ≥ (M (R) − 3C)/k = N (kR + kC + 3C) and thus the loop f (l) sits outside the ball B(f (x0 ), N (kR + kC + 3C)), and hence by assumption f (l) bounds a disk which does not intersect B(f (x0 ), kR+ kC + 3C). Let y1 , ..., yt be the vertices of the simplicial complex ϕ(D2 ) bounded by the loop f (l). The vertices f (x1 ), ..., f (xn ) are contained among the yj ’s. One can suppose that any triangle [yi , yj , ym ] of ϕ(D2 ) has edge length at most d. Therefore we have: d(g(yj ), xi ) ≤ d(g(yj ), gf (xi )) + C ≤ kd(yj , f (xi )) + 2C ≤ k 2 ε + (k + 2)C. This proves that xi , xj , g(ym ) span a simplex of Pa (H) (for all i, j, m) whenever we choose a larger than k 2 ε + (k + 2)C. Moreover: d(x0 , g(yi )) ≥ d(gf (x0 ), g(yi )) − C ≥

d(f (x0 ), yi ) − 3C − C ≥ R. k

Further there is a simplicial map ψ : ϕ(D2 ) → Pa (H) which sends f (xj ) into xj and all other vertices yk into the corresponding g(yk ). It is immediate now that ψϕ(D2 ) is a simplicial sub-complex bounded by l, which has the required properties. This proves the proposition 2.2.3. When G has torsion, one can construct a highly connected polyhedron with a free and cocompact G-action as follows (see [3]):

35

Definition 2.2.6. The colored Rips complex P (d, m, G) (for natural m) is the sub-complex of the m-fold join G ∗ G..... ∗ G consisting of those simplexes whose vertices are at distance at most d in G. Lemma 2.2.7. For m ≥ 3 and d large enough, G acts freely on the 2-skeleton of P (d, m, G) and π1 (P (d, m, G)) = 0. Proof. Clearly G acts freely on the vertices of Pd (G), hence any non-trivial g ∈ G fixing a simplex has to permute its vertices. Adding m ≥ 3 colors prevents therefore the action from having fixed simplexes of dimension less than 3. Now using the proof of lemma 2.2.1 one obtains also the simple connectivity. The quasi-isometry invariance of the simple connectivity at infinity follows now from the proof of the proposition 2.2.3, suitably adapted to the 2-skeleton of P (d, m, G). Moreover, the lemma 2.2.4 also implies the quasi-isometry invariance of the rate of vanishing of π1∞ . This ends the proof of the theorem 2.2.2. Remark 2.2.8. The same technique shows that the higher connectivity at infinity is also a quasi-isometry invariant of groups. Remark 2.2.9. The simple connectivity at infinity is not a quasi-isometry invariant for topological spaces (as the following example shows). Example: Consider X = (S 1 × R) 1∪ D2 S ×Z

and

Y = (S 1 × R)

∪

S 1 ×{0}

D2 .

Obviously π1 X = 0 = π1 Y and Y and X are two quasi-isometric spaces (the map which sends any disk D2 to its boundary is a quasi-isometry). However X and Y are not both sci: π1∞ X = 0 while π1∞ Y 6= 0. In fact, for any compact subset k ⊂ X, there exists another compact subset k ⊂ K ⊂ X such that every loop in X − K is null-homotopic in X − k (it is sufficient to take K = (S 1 × [−n, n]) 1 ∪ D2 with n sufficiently large). S ×[−n,n]

This is not true for Y , because if one takes k = D2 , then the null-homotopic loop γ = S 1 × {n}, (n 6= 0), is not null-homotopic in Y − K, for no compact subset K ⊇ k.

2.3

Uniform lattices in Lie groups

A lattice Γ in a real semisimple Lie group G is a discrete subgroup for which the quotient G/Γ supports a G-invariant measure of finite volume. One says 36

that Γ is uniform if this quotient is compact. The main source of lattices is the following: one constructs a periodic tiling of the corresponding symmetric space X = G/K, where K is a maximal compact subgroup of G, with one tile P either compact or of finite volume. The group of isometries of this tiling is then the required lattice. It seems very likely that uniform lattices in all connected Lie groups have a linear rate of vanishing of π1∞ (provided that the group is sci). Here we give a proof for semisimple, nilpotent and some solvable Lie groups. Observe that any uniform lattice in a Lie group G is quasi-isometric to G. Proposition 2.3.1. Uniform lattices in (non-compact) semi-simple Lie groups have linear rate of vanishing of π1∞ . Proof. We will denote below by dX the distance function and by BX the respective metric balls for the space X. Let G be a semi-simple Lie group. We can suppose that G has trivial center (otherwise we take G/Z). Any semi-simple Lie group with trivial center is a product of simple Lie groups. Hence we can suppose G simple. Let K be the maximal compact subgroup of the simple Lie group G and G/K the associated symmetric space. It is well-known that the Killing metric on G/K is non-positively curved, and hence the metric balls are diffeomorphic to standard balls, by the Hadamard theorem. If G is not SL(2, R) then K is different from S 1 and therefore it has e is compact. finite fundamental group. In particular the universal covering K The Iwasawa decomposition G = KAN yields a canonical diffeomorphism G → K × G/K. Furthermore we have a an induced canonical quasi-isometry e→K e × G/K. Large balls in G e can be therefore compared with products G e (compact) and metric balls in G/K, as follows: of K e e K×B e (r) ⊂ K×BG/K (a+r) ⊂ BG e (2a+r), for r large enough, G/K (r − a) ⊂ BG which implies our claim. ^R)) is quite similar, and a consequence of the (wellThe case of V1 (SL(2, ^R) and H 2 × R are canonically quasi-isometric. We known) fact that SL(2, ^R) will sketch a proof below. Let us outline the construction of the SL(2, geometry. A Riemannian metric on a manifold allows us to construct canonically a Riemannian metric on its tangent bundle, usually called the Sasaki metric. In particular one considers the restriction of the Sasaki metric to the unit tangent bundle U H 2 of the hyperbolic plane. Further there exists a natural diffeomorphism between U H 2 and P SL(2, R), which gives P SL(2, R) 37

a Riemannian metric, and hence induces a metric on its universal covering, ^R). This is the Riemannian structure describing the geometry namely SL(2, ^R). Observe that SL(2, ^R) and H 2 ×R are two metric structures on of SL(2, the same manifold, and both are Riemannian fibrations over H 2 (the former being metrically non-trivial while the later is trivial). It is clear now that the identity map between the manifolds U H 2 and ^R) H 2 × S 1 is a quasi-isometry, lifting to a quasi-isometry between SL(2, and H 2 × R. This implies that there are two constants a > 0, b such that: 1 2 dH 2 ×R (x, y) − b ≤ dSL(2,R) ^ (x, y) ≤ adH 2 ×R (x, y) + b, for all x, y ∈ H × R, a holds true. In particular we have the following inclusions between the respective metric balls: r 2 BH 2 ×R ⊂ BSL(2,R) ^ (r) ⊂ BH 2 ×R (cr) ⊂ BSL(2,R) ^ (c r), c for r large enough and c > 0. The claim follows. Remark 2.3.2. The same argument shows that the acyclicity radius for semisimple Lie groups is linear (see [38], section 4). The way to prove the claim for nilpotent and solvable groups consists in the large scale comparison with some other metrics, whose balls are known to be diffeomorphic to standard balls. While locally the Riemannian geometry of a nilpotent Lie group is Euclidean, globally it is similar to the CarnotCaratheodory non-isotropic geometry. Proposition 2.3.3. If G is a finitely generated nilpotent group then VG is linear. Proof. Since the torsion subgroup of a finitely generated nilpotent group is finite1 , one can suppose that G is torsion-free (up to quasi-isometry). It is known (see [48]) that a finitely generated torsion-free, nilpotent group G is a cocompact lattice in a real simply connected nilpotent Lie group GR , called the Malcev completion of G. We have also a nice characterization of the metric balls in real, nilpotent Lie groups given by Karidi (see [44]), as follows. Since GR is diffeomorphic to Rn it makes sense to talk about 1

the torsion elements of a nilpotent group form a subgroup; a subgroup of a finitely generated nilpotent group is finitely generated, and a finitely generated nilpotent torsion group is finite: hence the torsion subgroup of a f.g. nilpotent group is finite.

38

parallelepiped with respect to the usual Euclidean structure on Rn . Next, there exists some constant a > 0 (depending on the group GR and on the left invariant Riemannian structure chosen, but not on the radius r) such that the radius r-balls BGR (r) are sandwiched between two parallelepiped, which are homothetic at ratio a, so: Pr ⊂ BGR (r) ⊂ aPr ⊂ BGR (ar), for any r ≥ 1. This implies that we can take V1 (GR )(r) ∼ ar, and hence V1 (G) is linear. Remark 2.3.4. Metric balls in solvable Lie groups are quasi-isometric with those of discrete solvgroups, and so they are highly concave (see [28]): there exist pairs of points at distance c, sitting on the sphere of radius r, which cannot be connected by a path within the ball of radius r, unless its length is at least r0.9 . Further it can be shown that there are arbitrarily large metric balls which are not simply connected. Nevertheless we will prove that metric balls contain large slices of hyperbolic balls. Proposition 2.3.5. Cocompact lattices in solvable stabilizers of horospheres in product of symmetric spaces of rank at least two, or generic horospheres in products of rank one symmetric spaces have linear V1 . Proof. We give the proof for the most important solvable group, namely the 3-dimensional group Sol. It is well-known that Sol is isometric to a generic horosphere H in the product H 2 × H 2 of two hyperbolic planes (see the Appendix B at the end of this thesis for the definitions of horosphere, rank and distortion). Generic means here that the horosphere is associated to a geodesic ray which is neither vertical nor horizontal (or, equivalently, H is not entirely contained neither in the first H 2 nor in the second one). The argument in ([38] 3.D.), or its generalization from [23], shows that such horospheres H are undistorted in the ambient space i.e. there exists a ≥ 1, such that 1 dH 2 ×H 2 (x, y) ≤ dH (x, y) ≤ adH 2 ×H 2 (x, y), for all x, y ∈ H, a holds true. Here dH 2 ×H 2 and dH denote the distance functions in H 2 × H 2 and H, respectively. In particular we have the following inclusions between the respective metric balls: r H ∩ BH 2 ×H 2 ⊂ BSol (r) ⊂ H ∩ BH 2 ×H 2 (ar) ⊂ BSol (a2 r). a 39

Since the horoballs in H 2 × H 2 are convex it follows that the intersections H ∩ BH 2 ×H 2 (r) are diffeomorphic to standard balls. This proves that V1 (Sol) is linear. The linearity result extends without modifications to lattices in solvable stabilizers of generic horospheres in symmetric spaces of rank at least 2 (see [23]). Remark 2.3.6. It is known that finitely presented solvable groups are either simply connected at infinity or are of a very special form, as described in [50]. On the other hand it is a classical result that any simply connected solvable Lie group is diffeomorphic to the Euclidean space. It would be interesting to know whether a simply connected solvable Lie group can be isometrically embedded as a horosphere in a symmetric space. Remark 2.3.7. Notice that horospheres in hyperbolic spaces (and hence nongeneric horospheres in products of hyperbolic spaces) have exponential distortion, namely dH n (x, y) ∼ log dHn−1 (x, y), for x, y ∈ Hn−1 . This highly contrast with the higher rank and/or generic case. Hence one needs some extra arguments in order to extend the proof to all solvable Lie groups. Remark 2.3.8. One might notice a few similarities between V1 and the isodiametric function considered by Gersten (see [38]). Corollary 2.3.9. The rate of vanishing of π1∞ is linear for the fundamental groups of geometric 3-manifolds. Proof. There are eight geometries in the Thurston classification (see [66]): the sphere S 3 , S 2 × R, the Euclidean E 3 , the hyperbolic 3-space H 3 , H 2 × R, ^R), Nil and Sol. Manifolds covered by S 3 have finite fundamental SL(2, groups. Further the compact manifolds without boundary covered by S 2 × R are the two S 2 -bundles over S 1 , RP2 × S 1 or the connected sum RP3 ]RP3 , and the claim can be checked easily. As we already observed, this is the also the case for the Euclidean and hyperbolic geometries. The same holds for the product H 2 × R, in which case metric balls are diffeomorphic to standard balls. The remaining cases are covered by the previous propositions.

2.4

Non-uniform lattices

Let G be a connected, semisimple Lie group with trivial center and without compact factors. Unlike uniform lattices, nonuniform lattices Γ in G are not 40

quasi-isometric to the symmetric space X = G/K since they do not act cocompactly on X (for an extensive introduction see [64]). But one can consider the following construction: chop off every cusp of the quotient X/Γ and look at the lifts of each cusp to X, giving a Γ-equivariant union of horoballs in X. These horoballs are not disjoint in general; these can be made disjoint by cutting the cusps far enough out precisely when Γ has Q-rank one (see the Appendix B for precise definitions). The resulting space is called the neutered space X0 associated to Γ, and Γ acts co-compactly on it. We want to show that non-uniform Q-rank one lattices in higher rank Lie groups have a linear sci growth. The reason is that we want to use the deep result of Lubotzky–Mozes–Raghunathan ([45]) comparing the word metric of a non-uniform lattice with the induced metric of the symmetric space. Theorem 2.4.1. Let G be a simple Lie group of dimension at least 4 and of R-rank greater or equal to 2. Let Γ be an irreducible, non-uniform lattice in G of Q-rank one. Then Γ is sci with linear V1 . Proof. In order to prove that Γ has a trivial V1 , we have to look at balls in the Cayley 2-complex CΓ , endowed with the word metric of Γ. Actually, we shall compare metric balls of CΓ with metric balls of the neutered space X0 associated to Γ. Since Γ is irreducible and G has R-rank at least 2, then we can apply the Lubotzky–Mozes–Raghunathan’s Theorem (theorem B.0.36 of the appendix B), which says that the word metric of Γ is Lipschitz equivalent (hence quasiisometric) to the metric induced by G on Γ. Now, since Γ has Q-rank one, we have that Γ with the metric dG is quasi-isometric to the neutered space X0 with its length-metric (i.e. dG restricted to X0 ) because X0 /Γ is compact. Hence, by the quasi-isometry invariance of the sci growth, it will be sufficient to prove that X0 has a linear V1 . To do this we prove that the spheres in X0 are simply-connected (and hence V1 (X0 ) ≈ r). Let B0 (r) be a ball in X0 endowed with the metric dG . Any such a ball is contained in the ball BX (r) of X intersected with X0 . Since any ball in X is compact, and the neutered space is obtained from X where a collection of disjoint horoballs are deleted, then any ball of X intersects only finitely many horoballs. This implies that for any ball B0 (r) of X0 the following holds: ∂B0 (r) = ∂BX (r) − (∂BX (r) ∩ X0 ), namely, any sphere of radius r in X0 is obtained from a sphere of the symmetric space X, i.e. S n−1 , where a finite number of intersections of S n−1 with some horospheres are deleted. We need now a lemma: 41

Lemma 2.4.2. Let X be a proper CAT(0) space, H be a horoball, and B be a sphere of X. If the center c of B does not belong to H, then B ∩ H is convex (i.e. topologically a ball). Proof. Let fc be the distance function to c (fc (x) = d(x, c)), then fc restricted to H has only a critical point in H, namely the projection p(c) of c on H. There, it achieves a nondegenerate minimum. Since fc is proper, the level sets on H are balls (such balls retract on p(c)). Thanks to this lemma we obtain that the spheres in X0 are obtained from S n−1 where a finite number of disjoint disks Dn−1 are deleted (since the horospheres are disjoint). This means that, whenever the dimension n of X (i.e. of G) is ≥ 4, the spheres in X0 are simply connected. Thus V1 (X0 ) is linear. As already noticed, our Γ with the word metric is quasi-isometric to X0 with the metric induced from G; hence Γ has a linear V1 . Remark 2.4.3. We think the same result holds true for R-rank 1 Lie groups, and for non-uniform lattices of Q-rank > 1.

2.5

Coxeter groups

A Coxeter group is a group W with presentation of the following form hs1 , s2 , . . . , sn |s2i = 1 for i ∈ {1, 2, . . . , n}, (si sj )mij = 1i where i < j ranges over some subset of {1, 2, . . . , n} × {1, 2, . . . , n} and the coefficients mij belong to the set {2, 3, . . . , ∞}. Let W be a Coxeter group and C be its Cayley graph. We want analyze the asymptotic behavior of such a group. First, we need to recall two classical lemmas (see [49]). Lemma 2.5.1. If [v, w] is an edge of C then d(e, v) 6= d(e, w). Proof. Consider the homomorphism f : W → {+1, −1} defined by f (si ) = −1 for all i. Let p be an edge path joining the origin e with v. Then f (v) = (−1)n(p) where n(p) is the number of edges in p. If [v, w] is an edge in C from v to w, then d(e, w) = f (w) = (−1)n(p)+1 6= d(e, v). Lemma 2.5.2. Let v be a vertex of C. Suppose d(e, vsi ) = d(e, vsj ) > d(e, v) and mij 6= ∞. If α is a shortest path from 1 to v, then αh is geodesic when h is either one of the two edge paths of length mij , beginning at v alternating between edges with labels si and sj . 42

Proposition 2.5.3. Infinite Coxeter groups which are sci have linear sci. Proof. There are two ways for proving this proposition: as an application of a criterion developed by Davis and Meier in [20], and another using the semistability of Coxeter groups (Mihalik [51]). Below, we discuss both ways.

2.5.1

A proof using the semistability

Recall that a ray in X is a proper map r : [0, ∞) → X. Two rays r1 and r2 converge to the same end of X if for any compact C ⊂ X there exists an R such that r1 ([R, ∞)) and r2 ([R, ∞)) lie in the same component of X − C. The set of rays under this equivalence relation is the same as the set of ends of X. An end of X is semistable if any two rays of X converging to this end are properly homotopic. This is equivalent to the following: for any ray r converging to the end and for any n there exists a N ≥ n such that any loop based on r with image out of B(N ) can be pushed (rel. r) to infinity by an homotopy in X − B(n). Definition 2.5.4. Let X be a non-compact metric space, e an end of X and r a ray converging to e. The semistability function Se (n) is defined as the infimal N with the following property: for any R ≥ N and any loop l based on r which lies in X − B(N ) there exists a homotopy rel r sending l to a loop in X − B(R), the homotopy being supported within X − B(n). Notice that a semistable end has a well-defined semistability function. Set SG for SXe , where X is a compact space with fundamental group G, whenever this is defined. It is immediate that the growth of SG is independent on the space X and, using the same methods as before, one can also prove that SG is also a quasi-isometry invariant of the group G. Proposition 2.5.5. Assume that G is sci. Then V1 (G) ≤ SG . e is sci then for large N a loop within X e − B(N ) bounds Proof. If the space X a disk far from B(r). Use the semistability in order to homotope a loop lying e − B(SG (r)) to a loop within X e − B(N ), for N large enough. Hence any in X e − B(r), i.e. V1 (r) ≤ SG (r). loop out of B(SG (r)) is null-homotopic in X Proposition 2.5.6. Coxeter groups have linear semistability. Proof. We show that SG is linear. Let G = hs1 , . . . , sn |s2i = 1, (si sj )mij = 1i be the standard presentation of a Coxeter group, where mij ∈ {2, 3, ..., ∞}. 43

Lemma 2.5.7. Assume that mij 6= ∞ for all i, j. Then SG is linear. Proof. Let X be the standard 2-complex associated to this presentation. All e metric balls below are centered at the origin of the Cayley complex X. e Consider a loop l in X − B(r + 1). Let v be a vertex of l which realizes the minimum distance from the origin. The lemma 2.4.1 implies that two such vertices cannot be adjacent. Assume that the edges of l adjacent to v are [wv] labeled si and [v, u] labeled sj . Consider the null homotopic loop (si sj )mij which starts at v. This loop is made of wvu and its complementary wy1 ...yN u. By lemma 2.5.2 all points yj are at distance at least r+2 from the identity element. Further, to replace wvu by wy1 ...yN u it is sufficient a homotopy e − B(r). supported in X Continuing this procedure we can move the loop l as far as we wish. Lemma 2.5.8. Assume that SA and SB are linear, and C is finitely generated. Then SA∗C B is linear. Proof. For the proof of this lemma, we refer to the next chapter, where we study amalgamated free product. One can also check directly in [51]. Let G be a Coxeter group. If in its presentation there exist i, j such that mij = ∞, then one can split G into an amalgamated free product G = A∗C B where C is finitely generated and A and B have presentation with mij 6= ∞ (see [51]). Continued reduction of this fashion, and an application of the above lemmas yield the proof. Corollary 2.5.9. Simply connected at infinity Coxeter groups have linear sci growth.

2.5.2

A proof using Davis-Meier criterion

Given any finitely-generated Coxeter group W , there exists a contractible (actually CAT(0)) cell complex DW that W acts on cellularly, properly, and with finite quotient: the so-called Davis complex [18] (see also the appendix A below). The space DW may be cellutated so that the links of vertices of DW are all isomorphic to a fixed (finite) simplicial complex L, where L can be described combinatorially in term of subsets of the generating set of W . One can also run this construction backwards: given a flag complex (i.e. a simplicial complex such that any complete graph span a simplex) with I the vertex set, one gets a right-angled Coxeter group generated by si with i ∈ I 44

and with relations s2i = 1, (si sj )mij = 1 where mij = 1 if i = j, 2 if {i, j} span an edge, ∞ otherwise. It has been showed in [20] that a Coxeter group is sci if and only if its nerve L and all its punctured links L − σ are simply connected (for σ simplex of L). Consider then the union of several fundamental cells in the Davis complex. The boundary of such an union is a connected sum of various punctured links L − σ, and hence it is simply connected. Furthermore the metric balls in the complex are made of such unions of cells. Thus metric spheres are simply connected. The action of the Coxeter group on the Davis complex is not free but has finite stabilizers. Moreover there exists a finite index subgroup which acts freely on the Davis complex. The subgroup is still sci and by the previous arguments it has linear sci. This shows that the group itself has linear sci. Remark 2.5.10. The same idea works for the right-angled Artin groups. Given any flag complex L there is an associated Artin group AL (AL has generators in one-to-one correspondence with the vertices of L and defining relations stating that two generators commute when the corresponding vertices are adjacent in the 1-skeleton of L). One can then construct a K(AL , 1) whose universal covering is a contractible, CAT(0), cubical come L . Furthermore AL is sci if and only if the link of a vertex v ∈ K eL plex K is simply-connected. Hence a sci right-angled Artin group has a linear sci growth. Exotic local structure The above theorem says that, not only do the connectivity properties of the punctured links L − σ determine the connectivity at infinity of W (i.e. of DW ), but the converse is also true. This leads to speculation that, in the general context of nonpositive curvature, similar asymptotic-to-local results might hold. This is not the case as shown by M.Davis and J.Meyer: they construct in [21], by making minor modifications in the construction of DW , a model for a K(W, 1), write DW , with a CAT(0) cubical structure, but so that the connectivity properties at infinity do not descend to connectivity properties of links. Namely, W is sci, while the nerve of DW and its punctured links are not simply connected. This implies that for a sci group G, the closest property to the simple connectivity of large spheres in a K(G, 1), which is geometric (i.e. quasi-isometry invariant) is the rate of vanishing of the π1∞ . 45

2.5.3

Buildings

A building as originally defined by Tits is a certain combinatorial object. Associated to any building B there is a Coxeter group WB . In the classical examples of buildings WB is either finite (and the building is spherical ) or WB is a Euclidean reflection group (and the building is affine) . The geometric realization of a spherical or (irreducible) affine building is defined to be a certain simplicial complex, each top dimensional simplex being called a chamber. Embedded in this geometric realization there are many copies of the Coxeter complex of WB ; each copy is an apartment, and the building can be expressed as an union of apartments. In the case of spherical buildings each apartment is a sphere and in the (irreducible) affine case each apartment is a copy of the Euclidean space. Buildings associated to more general Coxeter groups arise in the theory of Kac-Moody groups as well as in the theory of graph products of groups, and are most easily defined in terms of chamber systems. As shown in [20] for the study of the end topology of buildings, it is more appropriate to define the geometric realization of a building B in such a fashion that each apartment is isomorphic to the Davis complex of WB , and so that the geometric realization of each chamber is a copy of L (the nerve of WB ), rather then a simplex. Because of the close connection between Coxeter groups and buildings, M.Davis and J.Meyer have extended the above result about the end topology of Coxeter groups to that of locally finite buildings. In particular they show that the geometric realization of a building is simply connected at infinity if and only if W is sci. It follows that: Corollary 2.5.11. Simply connected at infinity buildings have a linear sci growth.

2.6

The fundamental group at infinity

Before ending this chapter, we introduce some more complicated notions of the topology at infinity of spaces (and of groups): the semistability at infinity and the fundamental group at infinity for spaces which are semistable at infinity these definition first appeared in L.Siebenmann’s thesis [67]). We want just recall these notions to give some interesting relations between the simple connectivity at infinity of a space and the simple connectivity of its (visual) boundary. For convenience assume that a space X is a contractible, locally finite complex, and let Xn be a nested exhaustive sequence of compact subsets of 46

X. This exhaustive sequence induces inverse sequence of homotopy groups where, in order to keep track of basepoints, one chooses a proper ray r : [0, ∞) → X such that r([i, ∞)) ⊂ X − Xi for all i ∈ N: π1 (X − X0 , r(0)) ← π1 (X − X1 , r(1)) ← · · · π1 (X − Xn , r(n)) ← · · · where the morphism π1 (X − Xn , r(n)) ← π1 (X − Xn+1 , r(n + 1)) is defined via inclusion and the isomorphism π1 (X − Xn , r(n)) w π1 (X − Xn , r(n + 1)) induced by the path r([n, n + 1]). The inverse limit can depend on the proper homotopy class of the ray r (although there are no such examples for a finitely generated group). Such an inverse sequence is semistable if there is a function f : N → N such that the image of π1 (X − Xk , r(k)) in π1 (X − Xn , r(n)) is the same for all k ≥ f (n). In such a case, one can define the fundamental group at infinity, namely the inverse limit of this inverse sequence, and one writes π1∞ (X). A space is then sci if and only if it is semistable and the inverse limit is trivial. For a CAT(0)-space X, it is then natural to ask for relations between ∞ π1 X and π1 (∂X). The answer is far from being obvious and clear, since the semistability at infinity an the fundamental group at infinity are rather invariants of the so-called shape theory. However in [16] the authors give some interesting links between these two objects for CAT(0)-spaces. Lemma 2.6.1. For a CAT(0) space X, there is a natural homomorphism f from π1 (∂X) to π1∞ (X). This follows from the fact that there exists a well-defined retraction map from the sphere S(k) to S(k − 1) by x → [x0 , x] ∩ S(k − 1). Furthermore, in [16] one can find also: Proposition 2.6.2. If ∂X is one-dimensional then f is injective. The picture is optimal whenever ∂X admits a covering (i.e. it is locally compact), in fact: Theorem 2.6.3 (Conner–Fisher). If ∂X admits a universal covering space, then the natural homomorphism f : π1 (∂X) → π1∞ (X) is an isomorphism. Remark 2.6.4. Notice that there exist 1-ended CAT(0) groups that are simply connected at infinity, but have a boundary with non-trivial fundamental group. Such examples are 1-ended CAT(0) groups with non path-connected boundaries. 47

The first example is a group constructed by C.Croke and B.Kleiner in [17] which has more than one boundary. Croke and Kleiner observed that the right angled boundary of this group is not path connected. As this group is 1-ended, any of its boundaries are connected. The group in question has presentation H =< a; b; c; d : [a; b] = [b; c] = [c; d] = 1 >. The Cayley 2complex X of this presentation is a contractible 2-complex that is a union of planes any two of which have empty intersection or a line of intersection. If each square of X is given the geometry of [0; 1] × [0; 1], (the right angled geometry), then X becomes a CAT(0) space and H acts geometrically (discontinuously by isometries and cocompactly) on X. Croke- Kleiner observed that ∂X, the boundary of X, (with this geometry) is not path connected. As G is 1-ended, the direct product H × Z is simply connected at infinity. A boundary of H ×Z is the boundary of X ×R, which is simply the (unreduced) suspension of ∂X. (The unreduced suspension S(X), of a Hausdorff space X is the quotient space of X × [0; 1] with X × {1} and X × {0} identified to (separate) points). As ∂X is not path connected, a simple Mayer-Vietoris argument shows that the rank of the first homology of the suspension of ∂X is 1 less than the cardinality of the set of path components of ∂X. In particular, π1 (∂(H × Z)) is non-trivial.

48

Chapter 3 The end depth 3.1

On distinguishing one-ended groups

The simple connectivity at infinity and its refinement (the π1∞ growth) are “1dimensional” invariants at infinity for a group G, in the sense that they take care about loops and disks. The “0-dimensional” analogous of the simple connectivity (at infinity) is the connectivity at infinity, namely to be oneended. Hence, we can adapt the notion of the sci growth to a sort of growth of the end. Let us recall that a group G is one-ended (or 0-connected at infinity) if for any compact subset L of the Cayley graph C(G) of G, there exists a compact subset K ⊃ L such that any two points out of K can be joined by a path contained in C(G) − L. In order to measure the “kind” of connectivity at infinity of a one-ended metric space X, we introduce a functions measuring the “depth” of those connected components of X − B(r) which are bounded, as r → ∞. Definition 3.1.1. Let X be a one-ended metric space. Let B(r) be the r-ball of X, centered at the identity. The end-depth of X (or the growth rate of the connectivity at infinity), denoted V0 (X), is the infimal N (r) with the property that any two points which sit outside the ball B(N (r)) of radius N (r) can be joined by a path outside B(r). Remark 3.1.2. • For spaces which are k-connected at infinity, one can also consider the function Vk (X) = inf(N (r)) such that any k-sphere out of B(N (r)) bounds a (k + 1)-sphere outside B(r). 49

• It is easy to see that these functions can have arbitrary large growth for metric spaces (which are not Cayley’s complexes). In fact, let f : N → R+ be a function and let Ck (n) be the set ([0, f (n)] × S k )/R where R is the equivalence relation (0, x) ∼ (0, y) (i.e. Ck (n) is the cone of S k of height f (n)). Take the real half-line [0, ∞) and attach to any n the set Ck (n) at the cone point. The resulting space Xk is one-ended, kconnected at infinity, but the function Vk (n) is equal to n+f (n). Hence Vk (X) has the same growth as an arbitrary function f . • Observe also that the end depth V0 and the rate of vanishing of π1∞ , V1 , are not related in general. One can construct contractible manifolds with trivial V1 and a super-linear V0 ; it suffices to take a manifold such that the complementary of any ball B(r) is a disjoint union of simply connected components Ci becoming larger and larger as r growths. Similarly, there exist contractible manifolds with linear V0 and superlinear V1 ; something like a space with many “cylindric holes” becoming deeper and deeper. In this section we shall study the function V0 for groups. It is easy to see that the function itself depends on the presentation of a group, but the growth of the function (i.e. whether it is linear, polynomial or exponential) does not. Actually, the growth of the function V0 (that we shall continue to call end-depth) is a geometric property of groups. The aim of this section is to prove the following statement. Theorem 3.1.3. The growth rate of V0 is a well-defined quasi-isometry invariant of finitely presented groups. Observe that the function V0 depends on the presentation of a group. Let us recall a definition. An element g of a group G is a dead-end element with respect to a generating set S if it is not adjacent to an element further from the identity; that is, if a geodesic ray in the Cayley graph of (G, S) from the identity to g cannot be extended beyond g. The dead-end depth, with respect to S, of g ∈ G, is the distance in the word metric dS between g and the complement in G of the closed ball Bg of radius dS (1, g) centered at 1. If G − Bg is empty one defines the depth of g to be infinite. So g ∈ G is a dead end when its depth is at least 1. The next lemma indicates a relation between our function V0 and dead-ends elements. Lemma 3.1.4. Let P be a presentation for G. Let V0 be the end-depth with respect to P. If there exists r such that V0 (r) > r, then there exists a dead-element g in the sphere of radius V0 (r) of depth V0 (r) − r + 1. 50

Proof. Since V0 (r) > r, there exist x ∈ B(V0 (r)) and y ∈ X − B(V0 (r)) such that any path from x to y goes through B(r). To any such x one can associate a dead-end element in B(V0 (r)) in this way: if x is a dead-end element there is nothing to do, otherwise x is adjacent to some element x1 further from the identity. One can do the same for x1 and find x2 and so on. This process that has to stop after a finite number of steps, since the norm of xi cannot be greater then V0 (r), otherwise the point x could be joined with y outside B(r). Thus the set Dr ⊂ B(V0 (r)), consisting in all the dead-ends elements constructed in such a way, is non-empty. Let x¯ be the dead-end element in Dr with maximal norm. By construction of x¯, any point further from the identity can be joined with y by a path outside B(r). Hence the norm of x¯ is exactly V0 (r). Now, since any path joining x¯ with a point out of B(V0 (r)) goes into B(r), then the depth of x¯ is V0 (r) − r + 1. Notice that Cleary and Riley have constructed a group G such that G contains a sequence of dead-end elements gn at distance 4n from the identity, and of depth n with respect to one presentation, while G with respect to another presentation < S 0 , R0 > has dead-end depth bounded by 2 (see [14]). This implies that for a group, the property of having unbounded dead-end depth is not an invariant. Actually, they write that the behavior of the depth is hard to understand, for example the dead-end depth of Z is not uniformly bounded. On the other hand, to find groups with unbounded deadend depth seems to be an interesting problem; but if we wish something to be well-defined then we are forced to approach this problem from a different perspective. In order to define a group property which contains each of the above examples, let us introduce some definitions. Given two functions, f, g : N → N we say that f g if there are constant A, B, C such that f (n) ≤ An + Bg(Cn) We say that f and g are of the same type (and we write f ≈ g if f g and g f . Note that this is an equivalence relation. Using this definition we will say that G has a linear (or polynomial, exponential....) end-depth if (the equivalence class of ) V0 is. From now on, we will refer to V0 (or end-depth) to indicate the equivalence class of the function V0 . Our aim is to show that the equivalent class of V0 only depends on the group. Thus in some sense one provides here the right framework to study the connectivity at infinity. 51

Let G be a group and consider a compact complex X with G as fundae only depends on its mental group. We first observe that the end-depth of X 2-skeleton (actually the 1-skeleton). We can then restrict to 2-dimensional complexes with a given fundamental group. e be its universal Lemma 3.1.5. Suppose X is a finite two-complex. Let X e one-ended. Let T be a maximal tree in the 1-skeleton covering. Suppose X (1) e e Te. Then the end-depth of X e and Ye are equivalent. X of X and set Y = X/ Proof. Let C be the diameter of the (finite) maximal tree T of X. Let p e → Ye . Denote VX and VY the end depth of X e and the quotient map p : X Ye . Let BY (r) be the r-ball of Ye . We claim that two points a, b out of BY (VX (r + C) + C) can be joined by a path out of BY (r). In fact, the inverse e − BX (VX (r + C)). Hence image of such two points p−1 (a) and p−1 (b) sit in X e out of BX (r + C). The image p(α) is they can be joined by a path α of X still a path, joining a and b, which sits out of BY (r), since p collapse a tree of diameter C. The reverse implication is the same. Since finite 2-complexes with one vertex and with isomorphic fundamental groups are standard 2-complexes associated to two distinct presentations of the same group, for the end-depth to be a well-defined group property, we need to analyze how it change with respect to the presentation. Proposition 3.1.6. The end-depth of a group G does not change with the presentation. Proof. Let P and L two distinct (finite) presentations of the group G. One can pass from one to the other by an application of a sequence of Tietze transformations. Let us prove the result when P is gotten from L by applying a single transformation. Since the desired relation is clearly transitive, this suffices. Consider the different transformations: • (T1 ): add a new relator r, which is a consequence of the existing relators; • (T2 ): the inverse of (T1 ); • (T3 ): add a new generator y and a new relator of the form yu−1 , where u is an arbitrary word in the old generators; • (T4 ): the inverse of (T3 ). 52

We want to prove that these transformations does not change the growth of e1 be the Cayley 2-complex associated to L and X e2 the function V0 . Let X that of P . For transformations of type (T1 ) and (T2 ) it is obvious. Any edge path joining two points needs not to use the 2-cells. Consider now the transformations of type (T3 ) or (T4 ). It this case one f1 and X f2 . Let d1 be the metric on X f1 and d2 needs to compare the metrics of X f2 . Since the difference in the generating set is the presence be the metric of X of a new generator whose length in the other generating set is the norm of the word w, then we have the following inequalities: d1 ≤ d2 ≤ kwkd1 . Since f1 and X f2 are equivalent. kwk is constant, the end-depth of X An application of the previous two results yields the following corollaries: Corollary 3.1.7. Let K1 and K2 be finite connected complexes with isomorphic fundamental groups. Then V0 (K1 ) and V0 (K2 ) are equivalent. Corollary 3.1.8. Let G be a finitely presented group and H a subgroup of finite index. Then V0 (G) ≈ V0 (H). Proof. The results follows immediately from the fact that G and H have the same universal covering. Remark 3.1.9. This implies that the end-depth is a well-defined property of groups. Furthermore, the group of Cleary and Riley ([14]) belong to the class of groups with linear end-depth. Another interesting property of the end-depth is its quasi-isometry invariance. Proposition 3.1.10. The end depth is a quasi-isometry invariant for groups. Proof. The strategy of the proof is as follows. Denote by X and Y the Cayley 2-complexes associated to two quasi-isometric groups G, H respectively. Since G and H are quasi-isometric, there is a (λ, c)-quasi-isometry f : X → Y with quasi-inverse f 0 . Let VH be the end depth of H; in order to show that VG is equivalent to VH we use f to maps two points in X to two points in Y , we choose a path joining them in Y and map it back to X using the quasi-inverse f 0 ; a suitable approximation to the resulting (non-continuous) map of an interval to X yields an arc in X joining our original points. Let m = max{λ, c}, and BX (r) the ball of radius r in X. Let a, b be two points out of the ball BX (mVH (mr + m) + 4m) such that d(a, b) > m (otherwise 53

they are obviously joined by a path outside BX (r)). Thus the image points f (a) and f (b) are distinct and they sit outside the ball of radius VH (mr + m) (thanks to the property of f ). Then there exists a edge-loop l of length L in Y joining f (a) and f (b) out of BY (r). We can consider l as a map l : [0, L] → Y . Now, we associate to any point t ∈ [0, L] an element ht ∈ H such that either l(t) = ht or else ht is a vertex of the edge in which l(t) lies. Now define φ : {0, 1, 2, . . . L} → X such that φ(0) = a, φ(1) = b and φ(n) = f 0 (hn ) for n = 1, 2, . . . , L − 1. We claim that φ sends each pair of consecutive naturals to elements of G at distance at most 3m. Hence, one can send each edge [j, j + 1] to a geodesic in X joining φ(j) with φ(j + 1) outside B(r), provided the claim is satisfied. This will allow us to define a continuous map φ from an interval to X joining a and b and lying out of B(r), as wanted. We are left with d(φ(j), φ(j + 1)) ≤ 3m. If j and j + 1 are different from 0 0 and L, then d((φ(j), φ(j + 1)) = d(f 0 (hj ), fj+1 ) ≤ md(hj , hj+1 ) + m = 2m. In the case when j = 0 (or j + 1 = L), we have d(φ(0), φ(1)) = d(a, f 0 (h1 )) ≤ d(a, f 0 (f (a))) + d(f 0 (f (a)), f 0 (h1 )) ≤ m + (md(f (a), h1 ) + m) = m + 2m = 3m. Remark 3.1.11. The proofs of the quasi-isometry invariance of the enddepth and the sci growth can be used to show that the semistability function (defined in the previous section) SG for semistable at infinity groups is also a geometric property. Remark 3.1.12. For a one-ended, semistable at infinity group G, the enddepth V0 is bounded by the semistability function SG . e − B(SG (n)). Using the semistability, the conProof. Let x be a point in X stant loop x is homotope to a loop out of B(N ) for any large enough N , e − B(n). Thus x can be joined with a point x1 of with an homotopy in X e − B(N ) by a path in X e − B(n). Since G is one ended there exists N (n) X such that any two points out of B(N (n)) are joined out of B(n). Hence any e − B(n) are joined by the path [x, x1 ][x1 , y1 ][y1 , y] which sits out x, y ∈ X B(n). This means that V0 (G) ≤ SG . As already noticed, no example of group with super linear end-depth is known. However, most geometric examples of groups have a linear V0 , as the following statement shows. Proposition 3.1.13. If G is CAT (0) or hyperbolic (and 1-ended) then the end depth function V0 is linear. 54

Proof. Ontaneda in [54] proved that a proper cocompact CAT (0) space X is almost geodetically complete (for hyperbolic groups the same was proved by Mihalik, Bestvina-Mess). This means that there exists c > 0 such that for any x ∈ X there exists an infinite geodesics ray staring at 1 and which passes within c of x. Take two points x, y ∈ X − B(r + c); we have the rays rx , ry as above. Thus there are two points on the rays x0 ∈ rx , y 0 ∈ ry closed to x, y at distance < c. Moreover the points of the rays will eventually belong to the same connected component (because 1-ended) and could be joined by a faraway segment not intersecting B(r). Further we connect x to x0 by a segment of length < c and thus x0 is outside B(r); and x0 can be connected to y 0 by going along rays sufficiently far and then connecting the rays as above. This proves that we can take V0 (r) = r + c.

3.1.1

Amalgamated free product

In this section we will analyze the asymptotic behavior of amalgamated free products. It turns out that in this case the sci growth is linked with the end depth. Consider the standard amalgamated free product G = G1 ∗H G2 . Let X1 , YH and X2 be the standard 2-complexes associated to the presentations of G1 , H and G2 respectively. The space X obtained by attaching X1 and X2 e is constructed along YH has G as fundamental group. Its universal covering X from coset copies of the universal coverings of X1 and X2 which are attached e at coset copies of YeH (that we will denote by H). Proposition 3.1.14. If G1 , G2 are one-ended with linear V0 then the amalgamated free product G = G1 ∗H G2 has one end with a linear end depth. e The intersection of it with any Proof. Let B(r) be the ball of radius r in X. e1 or X e2 is compact, and then there exists a larger ball B(R) such copy of X e1 or X e2 in the complement of it are that any two points of any copy of X e − B(R) connected by a path out of B(r). Consider then two points a, b of X e1 or X e2 . Obviously these two points are not belonging to the same copy of X e connected by a path in the 1-skeleton of X, namely there exists an edge-path e1 or p1 , . . . , pt starting at a and ending at b with pi a path in one copy of X e2 . We proceed by induction on t. The case t = 1 is already proved. For X e Such an end point t > 1 one has that at least p1 (or pt ) ends in a copy of H. e (H is infinite) in such a way that it goes outside can be pushed along H 55

the ball B(R), and then it can be joined with a (or b) out of B(r). By the inductive hypothesis the proof ends. Theorem 3.1.15. If both G1 and G2 are simply connected at infinity with linear sci growth, and if H has one end with linear V0 , then the amalgamated free product G is simply connected at infinity with a linear V1 . Proof. Let B(r) be the r-ball of G centered at the identity. This ball intere1 and X e2 . Since X e1 and X e2 have linear sects a finite number of copies of X V1 there exists a R1 depending linearly on r so that any loop in one copy of e1 or X e2 out of B(R1 ) is contractible out of B(r). Since the subgroup H X e has one end and linear V0 , one can find a R2 such that in any copy of H, e any two points in the complement of B(R2 ) are connected by a path (of Y ) not intersecting B(R1 ). The proof that any loop out of B(R2 ) bounds a disk out of B(r) is now standard following [43]. The idea is that any edge loop starting at g ∈ G can be written in the form ga1 b2 a3 · · · bn , with ai ∈ G1 , bj ∈ G2 and a1 b2 a3 · · · bn = 1. This happens only if one of the ai ’s or bj ’s belongs to H (see [46]). Thus the path corresponding to such an element e out of B(R2 ), and so these two points starts and ends in the same copy of H e out of B(R1 ). The resulting loop is, then, can be connected by a path of H e1 or X e2 in the complement of B(R1 ). This means contained in one copy of X that it can be contracted out of B(r). The same reduction can be done with the resulting loop by induction on n. Remark 3.1.16. Note that it is only necessary that H be finitely generated since a presentation of G can be obtained using only the presentations of G1 and G2 . Remark 3.1.17. The same proof can be adapted to show that if G1 , G2 are k-connected at infinity with linear Vk , and if H is (k −1)-connected at infinity with linear Vk−1 , the G is k-connected at infinity with linear Vk . Theorem 3.1.18. If the subgroup H has one end with super-linear V0 and both G1 and G2 are simply connected at infinity, then G is a sci group with a super-linear V1 . e For any r there always exists Proof. Let B(r) be the ball of radius r of X. e e2 so that B(r) does not a coset copy of X1 intersecting a coset copy of X e1 and X e2 is intersect either one. Thus any loop contained in this copy of X e − B(r). But some larger ball B(R) intersects this copy of X e1 trivial in X e2 and divides the copy of H e in which they intersect into at least two and X 56

e −B(R) components. Since Gi are one-ended, one can construct a loop γ in X e1 connecting these two components which consists of a path in the copy of X e followed by a path in X e2 connecting the endpoints of the first path. In of H fact in the complement of any ball larger then B(R) having a radius within V0H (R), there is also such a loop connecting two points in the remaining parts e of the original components of H. 1 e e1 and M 2 = (X e − B(R)) ∩ X e2 . The Denote LR = (X − B(R)) ∩ X R e − B(R)) ∩ H e contains H e − BH (R). Thus one has intersection L1R ∩ MR2 = (X the following Mayer-Vietoris sequence : · · · → H1 (L) ⊕ H1 (M ) → H1 (L ∪ M ) → H0 (L ∩ M ) → · · · . Since the image of γ is not zero in H0 (L∩M ), then it is not zero in H1 (L∪M ), e − B(R)). So for large radius between R and V H (R) there and also in π1 (X 0 e − B(R)) but trivial in X e − B(r) exists a loop which is non-trivial in π1 (X with r < R. This implies that V1 for G is larger than linear. Remark 3.1.19. If G1 and G2 have one end and if the subgroup H has more then one end then G has one end but it is not simply connected at infinity. Remark 3.1.20. In the same way it can be shown that G has a superlinear Vk whenever G1 , G2 are k-connected at infinity with linear Vk and H is (k − 1)-connected at infinity with super-linear Vk−1 .

3.1.2

HNN-groups

Let G1 be a finitely presented group and let H be a subgroup. Let f be a monomorphism from H into G1 , and let K be the image f (H). Suppose that H is generated by a1 , . . . , an and denote by ci the generators f (ai ) of K. Let < b1 , . . . , bm , a1 , . . . , an , c1 , . . . , cn | p1 = 1, . . . , pk = 1 > be a presentation for G1 . Then the group G called the HNN-extension of G1 by f is the group with generators b1 , . . . , bm , a1 , . . . , an , c1 , . . . , cn , t with relations p1 = 1, . . . , pk = 1, c1 = t−1 a1 t, . . . , cn = t−1 an t. Consider the standard 2-complex G1 associated to G1 . It contains two subcomplexes H, K associated to H and K. Consider the space G obtained from a copy of G1 and a copy of H × [0, 1] where H × 0 is identified with the copy of H in G1 and H × 1 is identified with the copy of K in G1 . The e of G can be constructed from coset copies of G e1 universal covering space G e and H × [0, 1], the universal covering spaces of G1 and H.

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Proposition 3.1.21. If the group G1 has one end with linear V0 then the HNN-extension G has one end with linear V0 . Proof. By hypothesis, for any ball of radius r, any two points of G1 out of a ball of radius L(r) can be joined by a path out of B1 (r). Let B(r) be the e of radius r. Pick two points x, y in G e − B(r). The case when these ball of G e1 is already proved. Suppose then two points belong to the same copy of G e1 . The path p joining them that x and y belong to two different copies of G can be written as p = p1 t1 p2 t2 . . . pN tN , where pi are paths in the same copy e If this path does not intersect B(r), one has nothing to do. If not, of G1. there exists 1 ≤ n ≤ N such that pn intersect B(r) and pn−1 does not. The path pn can be pushed out of B(r) along the subgroup H in such a way that it does not intersect the ball. Doing this N − 1 times one obtains the case e1 . The proof is then achieved. when x, y are in the same copy of G Theorem 3.1.22. If the group G1 is simply connected at infinity with linear V1 and H has one end with linear V0 , then the HNN-extension G is sci with linear sci growth. Proof. The key tool is the Britton’s lemma: if g0 ti1 g1 ti · · · tin gn = 1 where gi ∈ G1 , then for some j, ij > 0, ij+1 < 0, and gj is in K or for some j, ij < 0, ij+1 > 0, and gj is in H. The proof is equivalent to the above one for amalgamations. e1 is sci with linear V1 then one Consider a ball B(r). Since each copy of G can enlarge (in a linear way) the ball in order to obtain a new ball B(L(r)) e1 bounds a such that any loop out of B(L(r)) contained in one copy of G disk not intersecting B(r). This new ball intersect a finite number of copies e × [0, 1]. Since H is one-ended, it is possible to find a larger ball B(R) of H e1 is for which any loop any loop out of it and contained in one copy of G e × [0, 1] out of contractible out of B(r), and any two points of one copy of H B(R) can be joined by a path in this copy not intersecting B(r). e − B(R). Any such a loop can be represented by Let γ be a edge-loop in G P i1 i in a word g0 t g1 t · · · t gn which is equal to 1 in G. If n |in | = 0 then the loop e1 contractible out of B(r). When P |in | > 0 is actually a loop in a copy of G n the Britton’s lemma can be used. The path tj gj tj+1 can be closed in H (or e1 that can be K) out of B(L(r)) in order to obtain a loop in a copy of G contracted out of B(r). Thus the loop γ is homotopic to a new loop with P n (|in |) two less. The proof ends by induction.

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Theorem 3.1.23. If the group G1 is simply connected at infinity and if the subgroup H has one end with a super-linear end-depth, then the HNNextension G is sci with super-linear V1 . Proof. The idea of the proof is the same as before. Since the subgroup H has a deep end, then we will be able to construct loops outside large balls bounding disks intersecting necessarily balls of small radius. e1 joined by a coset copy of For any r one can find two coset copies of G e × [0, 1], disjoint from B(r). At the same time there exists a larger radius H e1 and divides H e × [0, 1] which R1 for which B(R1 ) intersects these copies of G joins them into at least two components. One of these will be infinite, and another one will be deep, since the subgroup is one-ended with a super-linear e − B(R) one can always construct a V0 . Then for any radius R > R1 , in G loop γ as follows. Let (h1 , 0) and (h2 , 0) be two points of the two components e × [0, 1], and denote p a path of G e1 joining them. This path exists since of H G1 is one-ended. Such points can be chosen in such a way that h1 × [0, 1] and h2 × [0, 1] are in the complement of B(R). The loop γ is then composed of e1 joining p, followed by h2 × [0, 1], followed by a path in the second copy of G h2 × 1 with h1 × 1, followed by h1 × [0, 1] back to the starting point. Since the points h1 and h2 can be chosen further and further from B(R) since H has a super-linear V0 , one can always construct such a loop γ at a distance growing faster then a linear function from B(R), as R goes to infinity. One has then to observe that such a loop must be contracted within B(R). As before, a Mayer-Vietoris sequence shows that the loop γ cannot e − B(R). Denote L the parts of G e1 , H e × [0, 1/4) and be null-homotopic in G e × (1/8, 7/8) , then e − B(R) ∩ H e × (3/4, 1] in G e − B(R), and M = G H one has · · · → H1 (L) ⊕ H1 (M ) → H1 (L ∪ M ) → H0 (L ∩ M ) → · · · . The loop γ is not zero in (L∩M ), and so it cannot bound a disk out of B(R). Since this holds for any radius, we have that G has a super-linear V1 .

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Chapter 4 The weak geometric simple connectivity 4.1

Introduction

Casson and Poenaru ( [59], [34]) have developed an idea about the metric geometry of the Cayley graph of a group having to do with the covering conjecture in dimension three. The proof involves “approximating” the universal covering by compact, simply-connected three-manifolds. This condition was then adapted for groups by S. Brick in [6]. It seems very difficult (and plausibly undecidable) to check whether a given noncompact polyhedron (or complex) verifies the qsf condition of Brick. We consider a related and apparently stronger notion which is called below the wgsc. We prove that the two notions are actually equivalent for discrete groups i.e. for the universal coverings. The main interest of such a strengthening is the hope that it might be somewhat easier to prove that specific high dimensional complexes are not wgsc, due to a criterion developed in [30]. Moreover, there exist uncountably many open contractible manifolds for any n 6= 4 which are not wgsc ([30]). This would suggest that non wgsc Cayley complexes (i.e. discrete groups) should also exist. The problem of finding aspherical non wgsc manifolds which are universal coverings seems to be much more difficult. Here we want to study the wgsc property for groups, by showing that it is an interesting group-theoretic notion. This agrees with the idea that killing 1-handles of manifolds is a group-theoretical problem in topological disguise.

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4.2

Definition of wgsc groups

The following definition due to C.T.C.Wall came out from the work of S.Smale on the Poincar´e Conjecture and, more recently, in the work of V.Poenaru ([59]). Moreover, it has been revealed as especially interesting in the noncompact situation, in connection with uniformization problems (see [30]). We will focus in the sequel to non-compact manifolds, which will mostly be universal coverings of compact polyhedra: Definition 4.2.1. A non-compact manifold (possibly with non-empty boundary), is geometrically simply connected (abbreviated gsc) if there exists a proper handlebody decomposition without 1-handles. Remark 4.2.2. Handle decomposition are known to exist for all manifolds in the topological, PL and smooth settings, except in the case of non-smoothable topological 4-manifolds. Notice that open 4-manifolds are smoothable. Manifolds and handlebodies considered below are PL. One has the following combinatorial analog of the gsc for polyhedra: Definition 4.2.3. A non-compact polyhedron P is weakly geometrically simply connected (abbreviated wgsc) if P = ∪∞ j=1 Kj , where K1 ⊂ K2 ⊂ ... ⊂ Kj ⊂ ... is an exhaustion by compact connected sub-polyhedra with π1 (Kj ) = 0. Alternatively, any compact subspace is contained in a simply connected sub-polyhedron. Remark 4.2.4. Notice that a wgsc polyhedron is simply connected. The wgsc spaces with which we will be concerned in the sequel are usually polyhedra. Similar definitions can be given in the case of topological (respectively smooth) manifolds where we require the exhaustions to be by topological (respectively smooth) submanifolds. Remark 4.2.5. The two concepts from above are closely related. In fact, the noncompact manifold W n , (n 6= 4), which one supposes to be irreducible if n = 3, is wgsc if and only if it is gsc. (For n ≥ 5 this is a consequence of Wall’s result stating the equivalence of gsc and simple connectivity in the compact case; if one has an exhaustion by compact, simply connected submanifolds Ki , then Ki are gsc and one can glue together these intermediary decompositions). Definition 4.2.6. The finitely presented group Γ is wgsc if there exists some e is compact polyhedron X with π1 (X) = Γ such that its universal covering X wgsc. 61

Figure 4.1: A 2-complex associated to a presentation of Z. Remark 4.2.7. Working with complexes instead of polyhedra in the definitions above, thus not allowing subdivisions, yields the same notion of wgsc. One problem with the wgsc is that it cannot be read directly on an arbitrary complex of given fundamental group. In fact: Proposition 4.2.8. 1 For any finitely presented group Γ with an element of infinite order, there exists a complex X with π1 X = Γ whose universal covering is not wgsc. Proof. We begin by showing that the proposition is true for the integers group Z. The standard complex associated to Z is a circle S 1 (i.e. a loop a based at x). If we add two new generators b and c (attached at v), with the following relations c = aba−1 and bcb−1 = 1, we obtain another presentation of Z. This is equivalent to the presentation ha, b|baba−1 b−1 = 1i. The standard two complex associated to this presentation is constructed as follows. Consider a wedge of two circles a and b with base point x and a cone C = (S 1 , e) × [0, 1]/ ∼, where ∼ identify (S 1 , e) × {1} with a single point. Now, consider the segment I joining (e, {0}) with (e, {1}). Finally, attach the base circle of C with b (in such a way that e ≡ x), and identify the 1

We are grateful to F.Lasheras for this proposition.

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Figure 4.2: The universal cover is not wgsc.

segment I with the loop ab. This is the compact 2-complex X associated to the above presentation of Z (in this complex the loop b is homotope to zero). The important observation is that its universal covering is not wgsc. The e is constructed from the universal cover of a (i.e. the Cayley reason is that X graph of Z), with a copy of b, say bi , attached to each xi ∈ {p−1 (x)} and a copy of C, say Ci , attached to bi with I identified with the edge [pi , pi+1 ]bi+1 . This space is not wgsc, since any loop bi is killed by Ci , but such a Ci creates a new loop bi+1 , which is killed by Ci+1 , and so on. However, observe that this space is still qsf (and also sci). Now, let Γ be a finitely presented group with an element of infinite order. Then there exists a generator a of Γ of infinite order (a finitely presented torsion group is finite). Then one can add to a presentation of Γ a new generator b and a new relation as before. Then in the universal covering the only way to kill b is along the cones Cj . Hence it is not wgsc. Let P be a finite presentation of a finitely presented group G. Let X be the standard 2-complex corresponding to the presentation P. Any finite 2-complex with only one vertex can be seen as a standard complex of some e of X is the finitely presented group. The 1-skeleton of the universal cover X Cayley graph of G.

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The property wgsc cannot be read directly on an arbitrary complex, but only after stabilizing. Let us recall some definitions: Definition 4.2.9. Let X be a 2-dimensional complex and let dn , en+1 ⊂ X be cells of X, d being a free face of e when considered as a subcomplex of X. An elementary internal collapse (e, d) in X consists of “collapsing” the cell e through its face d, even if d is not a free face of e within the entire complex X. Notice that this operation produce a new complex proper homotopy equivalent to X. Any of the inverses of an elementary internal collapse is called an elementary internal expansion. A stabilization consists of adding one additional 2-cell along the boundary of an union of existing 2-cells (and thus preserving the fundamental group). Proposition 4.2.10. Let X and Y be finite complexes having isomorphic e is wgsc then Ye 0 is wgsc, where Y 0 is obtained from fundamental groups. If X Y by finitely many internal expansion and stabilizations. Proof. Remark first that the wgsc is a property related to the 2-skeleton. In fact, we have: Lemma 4.2.11. The complex Z is wgsc if and only if its two dimensional skeleton ske(2) (Z) is wgsc. Proof. Let C be a compact subset of ske(2) (Z). If the complex Z is wgsc, then C is contained in some finite, simply-connected sub-complex L of Z. Then, the restriction to the 2-skeleton of L is a finite simply-connected sub-complex of ske(2) (Z) containing C. For the other direction, let K be a compact subset of Z. Then K is contained in a finite sub-complex C of Z. Since ske(2) (Z) is wgsc, the 2-skeleton of C is contained in a finite simply-connected subcomplex L of ske(2) (Z). Now, consider the complex obtained by adding to L the cells of dimensions ≥ 3 of C. Since this does not affect being simplyconnected, we have a finite, simply-connected sub-complex of Z containing K, as wanted. Lemma 4.2.12. Let X be a finite 2-complex. Let T be a maximal tree of the 1-skeleton of X and denote by Y = X/T the complex in which T is identified e is wgsc whenever Ye is wgsc, while if X e is wgsc then to a point. Then X the 2-complex obtained by Ye by a (possibly infinite) sequence of elementary internal collapses and/or expansions is wgsc. e → Ye . Proof. Let p the quotient map X

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e Observe that for Suppose that Ye is wgsc. Let C be a finite subcomplex of X. e obtained any g ∈ π1 (X) there exists a copy Tg of the maximal tree T in X by left translation by g from a given lift. Now, C intersects only finitely many such copies Tg , since it is compact. Let CT be the union of C with all the Tg ’s which intersect C. One knows that p(CT ) is contained into some finite simply connected subcomplex K of Ye . Then the subcomplex p−1 (K) is compact and simply connected, and it contains C. e is wgsc we have the following problem: if K is a compact, Whenever X e its image p(K) may not be simply-connected simply-connected subset of X, (for example if one of the Tg ’s is not entirely contained in K and its intersection with K is not connected). Hence we need to change a little bit the complex to avoid this kind of problems, thanks to the following lemma from [13]: Lemma 4.2.13. Let X be a 2-dimensional wgsc complex and let Ti , i ∈ I, be a (locally finite) collection of trees inside X. We can get a new wgsc 2-complex X 0 (which is proper homotopy equivalent to X), obtained from X by a (possibly infinite) sequence of elementary internal collapses and/or expansions still containing the Ti ’s and with an exhaustion (Cn ) (i.e. X 0 = ∞ ∪ Ci , with Cj ⊂ Cj+1 ) by compact, simply connected sub-complexes such i=1

that each intersection Cn ∩ Ti (n ≥ 1, subtree (and hence contractible).

i ∈ I) is either empty or a connected

e is wgsc. The With this lemma we can finish up. Assume now that X 0 previous lemma builds a new complex X which is also wgsc. Denote by Y 0 the complex obtained by X 0 in which T is identified to a point. Let B be e 0 , and a compact subset of Ye 0 . Then A = p−1 (B) is a compact subset of X so there exists a simply connected, finite subcomplex K containing A. In this case, the compact subcomplex p(K) is simply connected (thanks to the e 0 ) and it contains B. Hence Ye 0 is wgsc. property of the filtration of X Lemma 4.2.14. Let X1 and X2 be two finite 2-complexes with one vertex f1 is wgsc if and only the each and isomorphic fundamental groups. Then X polyhedron Y obtained from X2 by stabilizations has a wgsc universal cover Ye . Proof. The standard 2-complex corresponding to the presentation of a group has one vertex v, one loop for each generator and 2-cells attached to loops corresponding to the relators. Any finite 2-complex with only one vertex is

65

the standard complex associated to some finite presentation of its fundamene of X is the Cayley graph tal group. The 1-skeleton of the universal cover X of G. Thus the two complexes Xi correspond to two presentations P1 and P2 of their fundamental group. It is then a classical result that one can obtain P2 from P1 by using a finite sequence of Tietze transformations: • (T1 ): add a new relator r, which is a consequence of the existing relators; • (T2 ): the inverse of (T1 ); • (T3 ): add a new generator y and a new relator of the form yu−1 , where u is an arbitrary word in the old generators; • (T4 ): the inverse of (T3 ). We want to see how an application of a Tietze transformation affects the f1 is wgsc and thus the ascending union of finite wgsc condition. Assume X simply connected complexes Ki . f2 is obtained from X f1 by attaching one extra 2-cell at each 1. (T1 ). X e j is the union of vertex along the path labeled by the relator r. If K e j is Kj with all extra 2-cells whose boundary is contained in Kj then K f2 is wgsc. also finite and simply connected, hence X f1 , let uv be the edge path with initial 2. (T3 ). For each vertex v ∈ X f2 is obtained from point v which is determined by the word u. Then X f1 by attaching a new edge ev and then a 2-cell Dv at each vertex X f1 , such that the initial point of ev is v, the endpoint of ev is the v∈X endpoint of uv , and Dv is attached along the path ev u−1 v . Consider f f a compact subset A of X2 . By hypothesis, A ∪ X1 is contained in a finite subcomplex K which is simply connected. Now, A intersects only finitely many Dv ’s, say Dvi , i = 1, n. Then K ∪ Dvi contains A and it is simply connected. f1 and X f2 are reversed. In particular the edges ev 3. (T4 ). The roles of X f1 . The edges ev are free and we can collapse and the 2-cells Dv lie in X each Dv onto the path uv . The images of the compacts Ki are then simply connected compacts in X2 . 66

However the transformation (T2 ) does not necessarily preserve the wgsc. We f2 ⊂ X f1 , is obtained by deleting one 2-cell D2 on each vertex, know that X f2 is still simply connected. which corresponds to r. It is not clear that Kj ∩ X In fact a loop in Kj is null-homotopic and if a null-homotopy uses a sliding through the extra D2 then we have to replace it by the homotopies along the constituents of r. However, the later ones might well go out of the complex Kj . However, there exists a slight reformulation of the Tietze theorem as follows. Let consider two presentations P1 and P2 of a group. There exist then two sequences of Tietze moves consisting only in the transformations (T1 ), (T3 ) and (T4 ) which change Pi into a common presentation Q. Moreover the complex associated to Q can be gotten by stabilizations from that of the Pi . This proves the claim. One can also prove the following: Proposition 4.2.15. Let G be a wgsc group and X be any compact 2polyhedron with π1 (X) ∼ = G. Then, the universal cover of X ∨ S 2 has the proper homotopy type of a wgsc complex. Proof. Let G be a wgsc group and Y be a compact 2-polyhedron with π1 (Y ) ∼ = G and whose universal cover is wgsc. Let X be any other compact 2polyhedronWwith π1 (X)W∼ exist finite = G. By ([73], Thm. 14), W there W bouquets 2 2 2 of spheres i∈I S and j∈J S such that X ∨( i∈I S ) and Y ∨( j∈J S 2 ) are homotopy equivalent, and hence their covers are proper homotopy W universal 2 equivalent. Assume the bouquet i∈I S is being attached to X through a ˜ → X be the universal covering projection. Let vertex v ∈ X, and let p : X ˜ be a tree containing all the vertices in p−1 (v). Finally, according to T ⊂X the classification of spherical objects under T described in ([2], Prop. 4.5(b)) and the corresponding Gluing W Lemma in proper homotopy theory (see [2]), the universal cover of X ∨ ( i∈I S 2 ) is proper homotopy equivalent to the ˜ by attaching only one sphere S 2 at a time through space obtained from X every vertex in p−1 (v). The latter space is the universal cover of X ∨ S 2 , and hence it is proper homotopy equivalent to Y^ ∨ S 2 . Moreover Y^ ∨ S 2 is wgsc since Ye was supposed wgsc, and the conclusion follows. Notice that one does not know whether the wgsc is a proper homotopy invariant for general complexes (even if it was proved in [30] that it does when restricting to manifolds). 67

Then, the wgsc condition might not be satisfied for all polyhedra with given fundamental group, and it rather seems to be only an invariant of a group presentation. Remark 4.2.16. We further remark that the wgsc property cannot be extended for finitely generated groups (having infinitely many relations), since for any such a group there exists a presentation with infinitely many relations such that the group is wgsc with respect to this presentation. To do this, it suffices to add infinitely many 2-cells, along the boundary of an union of existing 2-cells, killing the fundamental group of any compact subset. Remark 4.2.17. Recall that there exist uncountably many (Whitehead-type) manifolds which are not wgsc ([30]), but these manifolds are not covering spaces (i.e. they admit no discrete cocompact group of transformation). Remark 4.2.18. In higher dimension (≥ 5) a non wgsc manifold has to be non-simply connected at infinity ( by [61]). Furthermore, a necessary condition for a manifold to be not wgsc is that for any exhaustion Ki by compact subsets and for any i, there exists N = N (i) such that KN is not null-homotopic outside a small ball at the identity.

4.2.1

The qsf property after Brick and Mihalik

The qsf property is a weaker version of the wgsc, which has the advantage to be independent on the polyhedron we chose. Specifically, Brick ([6]) defined it as follows: Definition 4.2.19. The simply connected non-compact PL space X is qsf if for any compact C ⊂ X there exists a simply connected compact polyhedron K and a PL map f : K → X so that C ⊂ f (K) and f |f −1 (C) : f −1 (C) → C is a PL homeomorphism. Definition 4.2.20. The finitely presented group Γ is qsf if there exists a compact polyhedron P of fundamental group Γ so that its universal covering Pe is qsf. Remark 4.2.21. It is known (see [6]) that the qsf is a group property and does not depend on the compact polyhedron P we chose in the definition above. In fact, if Q is any compact polyhedron of fundamental group Γ then e is also qsf. Q

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Remark 4.2.22. The qsf is very close to another concept developed by Poenaru in [59], and then considered in [29] and [30]: the Dehn-exhaustibility (defined below). Poenaru proved a Dehn-type lemma, which states that a Dehn-exhaustible 3-manifold is wgsc (see [59]). This result is purely 3dimensional and cannot be extended in higher dimension. We will show that the Dehn-exhaustibility and the qsf property are actually equivalent for groups. Definition 4.2.23. The simply-connected n-manifold W n is Dehn-exhaustible if, for any compact K ⊂ W n , there exists some simply connected compact polyhedron L and a commutative diagram f K −→ L i& .g Wn where i is the inclusion, f is an embedding, g is an immersion, and f (K) ∩ M2 (g) = ∅. Here M2 (g) is the set of double points, namely M2 (g) = {x ∈ L : |g −1 g(x)| ≥ 2} ⊂ L. If n = 3, then it is required that the map g be a generic immersion, which means here that it has no triple points.

4.2.2

Small depth and 1-tame groups

Now we consider some other tameness conditions on non-compact spaces, which are closely related to the wgsc. Moreover we will show later that they induce equivalent notions for discrete groups. Definition 4.2.24. The simply connected non-compact PL space X has small depth if for any compact C ⊂ X there exist two compact sub-polyhedra C ⊂ D ⊂ E ⊂ X, fulfilling the following properties: 1. The map π1 (D) → π1 (E) induced by the inclusion, is zero. 2. Any loop in E − C (based to a point in D − C) is homotopic rel the base point within X − C to a loop which lies entirely inside D − C. Alternatively, let us denote by ιY : π1 (Y − C) → π1 (X − C) the morphism induced by inclusion (for any compact Y containing D), by fixing a base point (which is considered to be in D − C). Then one requires that ιD (π1 (D − C)) = ιE (π1 (E − C)). The finitely presented group Γ has small depth if there exists a compact polyhedron P of fundamental group Γ so that its universal covering Pe has small depth. 69

Remark 4.2.25. An obvious variation would be to ask that the homotopy above might not keep fixed the base point. We don’t know whether the new definition is equivalent to the former one. Definition 4.2.26. The space X is 1-tame if any compact sub-polyhedron C is contained in a compact K ⊂ X, so that any loop γ in K is homotopic within K to a loop γ 0 in K − C, while γ 0 is null-homotopic within X − C. The finitely presented group Γ is 1-tame if there exists a compact polyhedron P of fundamental group Γ so that its universal covering Pe is 1-tame. Remark 4.2.27. Notice that one does not require that an arbitrary loop in K − C be null-homotopic within X − C. This happens only after a suitable homotopy which takes place in K. Proposition 4.2.28. Both the small depth and 1-tameness can be welldefined for groups. Proof. We shall prove that these conditions are equivalent to the qsf condition, which is independent on the choice of the polyhedron. If one proves directly the invariance under Tietze moves, one has to admit stabilizations (again because of transformation of type (T2 )).

4.2.3

Comparison of the various tameness conditions

Proposition 4.2.29. A wgsc polyhedron has small depth and is 1-tame. A polyhedron which is either 1-tame or else has small depth is qsf. Proof. Let C be a compact sub-polyhedron of the polyhedron X. (1). Assume that X is wgsc. Then one can embed C in a compact 1connected sub-polyhedron K ⊂ X. Take then D = E = K, to prove that X has small depth. Further K is convenient for showing that X is 1-tame. (2). Suppose that X has small depth, and D and E are the sub-polyhedra provided by the definition above. Any loop γ in E, based at a point in C is a connected sum of the loops γ 0 ⊂ D, and γ j ⊂ E − C. Moreover γ j are disjoint for j ≥ 1 and γ j ∩ γ 0 are connected arcs. Further, by hypothesis γ 0 is null-homotopic in E. Next γ j is homotopic within X − C to a loop γ¯j , which can be arranged to have a common point with γ j . Assume now that we chose a system of generators γ1 , . . . , γn of π1 (E). We will do the construction above for each loop γj , obtaining the loops γj0 nullhomotopic in E, and the loops γjk in E−C which are homotopic to γ¯jk in D−C. 70

b by adding 2-disks along the composition of We define first a polyhedron E k k the loops γj γ¯j . Recall that these two loops have a common points, and so it makes sense to consider their composition. b → X, which extends the inclusion There is defined a natural map F : E E ,→ X, as follows. For each j, k, fix the base point to be the common point of γjk and γ¯jk . There exists a homotopy within X − C keeping fixed the base point of γjk between γjk and γ¯jk . Alternatively, there exists a free null-homotopy of the loop γjk γ¯jk within X − C. We send then the 2-disk of Eˆ capping off γjk γ¯jk onto the image of the associated null-homotopy. It is clear than F is a homeomorphism over C, since the images of the extra b is simply connected. 2-disks are disjoint from C. Moreover, we claim that E b is homotopic to a loop within E, and hence to a In fact, any loop in E composition of γj . Each γj0 is null-homotopic in E since it lies in D. Each γjk b to γ¯jk which lies also in D −C, and hence is null-homotopic is homotopic in E b = 0. in E. Therefore π1 (E) (3). Suppose now that X is 1-tame. Let K be the compact associated to an arbitrarily given compact C. Any loop γ in K is freely homotopic to a loop γ¯ in K − C. Consider γ1 , · · · , γn a system of generators of π1 (K). b by adding 2-disks along the loops γ¯j . There We construct the polyhedron K b → X, which extends the inclusion K ,→ X, defined as exists a map F : K follows. The 2-disk capping off the loop γ¯j is sent into the null-homotopy of γ¯j b within X −C. Then F is obviously a homeomorphism over C. Meanwhile, K is simply connected since we killed all conjugacy classes of loops from K. Proposition 4.2.30. If the n-manifold M n is qsf and n ≥ 5 then M n is gsc. Proof. The main ingredient is the following lemma: Lemma 4.2.31. If the manifold M n is qsf, then the manifold M n × Dn+1 is gsc, where Dk stands for the closed ball of dimension k. Proof. It suffices to consider compacts within M n × Dn+1 , of the form C × Dn+1 , where C is an arbitrary compact C ⊂ M n . Further, it is enough to consider that C is a codimension zero sub-manifold of M n , since any compact is contained in a such sub-manifold. By hypothesis there exists a simply connected polyhedron K and a PL map f : K → M n which is a homeomorphism over C. We denote by C¯ the preimage f −1 (C). We identify next M n with M n × {0} ⊂ M n × Dn+1 , where Dk stands for the closed ball of dimension k. By composing with the inclusion we obtain a map 71

¯ → F : K → M n × Dn+1 . Let us consider the induced map F : K − int(C) n n+1 (M −int(C))×D , which is an embedding when restricted to the boundary ¯ ∂ C. By general position arguments there exists an isotopy of F which is fixed on ¯ → (M n − int(C)) × (a regular neighborhood of) ∂C to a map G : K − int(C) Dn+1 , which is an embedding. ¯ be a regular neighborhood of G(K − int(C)) within Let Θ(K − int(C)) (M n − int(C)) × Dn+1 . Then the boundary of Θ(K − int(C)) is a regu¯ and also a regular neighborhood lar neighborhood of ∂ C¯ in Θ(K − int(C)), of ∂C in (M n − int(C)) × Dn+1 , and hence homeomorphic to ∂C × Dn+1 . ¯ and a copy of C ×D1n+1 in order to obtain We can glue together Θ(K −int(C)) a regular neighborhood Θ(K) of K inside M n × Dn+1 . Here D1n+1 is a concentric ball laying inside Dn+1 . Furthermore there exists a homeomorphism supported on a neighborhood of C × Dn+1 which transforms C × D1n+1 into C ×Dn+1 . This way one transforms Θ(K) into another regular neighborhood containing C × Dn+1 . Remark that Θ(K) is simply connected since it collapses onto K. This proves that M n × Dn+1 is gsc. The next step uses the invariance of the gsc at proper homotopy, under the assumption n ≥ 5, according to [30], Theorem 6.3. This implies that M n must be gsc, as claimed. Remark 4.2.32. More generally, if P is a finite dimensional polyhedron which is qsf then P × Dn is wgsc, for large enough n. Remark 4.2.33. A similar result holds for Dehn exhaustibility. In particular a n-manifold which is Dehn-exhaustible is wgsc provided n ≥ 5, and for n 6= 4 a wgsc n- manifold is Dehn-exhaustible. Corollary 4.2.34. The wgsc, qsf, Dehn-exhaustibility, 1-tameness and small depth are equivalent for finitely presented groups. This also yields the following characterization of the wgsc: Corollary 4.2.35. The group Γ is wgsc if and only if the universal covering fn of any compact manifold M n with π1 (M n ) = Γ and dimension n ≥ 5 is M wgsc (or gsc). In particular, a qsf group admits a presentation whose Cayley graph is wgsc.

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4.2.4

The wgsc growth

Let P be a finite presentation of the wgsc group G and C(G, P ) be the associated Cayley complex, endowed with the natural word metric. Denote by B(r) the metric ball of radius r centered at the identity. A π1 -resolution of the polyhedron C inside C 0 is a PL-map f : A → C 0 such that f : f −1 (C) → C is a homeomorphism, where π1 (A) = 0. We want to refine the wgsc by introducing the wgsc growth function fG : fG (r) = inf{R such that there exists a π1 −resolution of B(r) into B(R)}. One can show easily that the rough equivalence class of fG (r) depends only on the group G and not on the particular presentation, following [6]. Remark 4.2.36. Note that π1 (B(r)) → π1 (B(fG (r))) is zero. Thus fG is bounded from below by the connectivity radius R1 introduced by Gromov in y [38]. The group Γ =< x, y|xx = x2 > due to Gersten has connectivity radius (and thus wgsc growth) larger than any iterated exponential (see [38], 4.C). Notice that this groups is wgsc since is a 1-relator group. Remark 4.2.37. Observe that a wgsc group whose wgsc growth fG is recursive has a solvable word problem. Proof. The only thing to remark is that the growth rate of the wgsc is an upper bound for the Gersten isodiametric function, and the word problem is solvable whenever the isodiametric function is recursive. Indeed if a word w is 1 in G =< S, R >, then it is a product of relators of the form uru−1 . The isodiametric function f allows to choose these conjugates in a such a way that |u| ≤ f (|w|), that is equivalent to say that w = 1 ⇐⇒ w ∈ H where H is a subgroup of the free group FS generated by uru−1 such that r ∈ R and |u| < f (|w|). These elements are a finite number and hence H is of finite type. The problem to decide whenever a word of a free group belongs or not to a finite type subgroup is solvable. Then one has an algorithm to decide if w = 1 or not. Remark 4.2.38. There exists a finitely presented wgsc group which has unsolvable word problem. Indeed in [15] the authors construct a group with unsolvable word problem that can be obtained from a free group by applying three successive HNN-extensions with finitely generated free associated subgroups (such a group is wgsc from (2) of the next corollary). Remark 4.2.39. The problem of finding a finitely presented group which is not wgsc seems not easy. Further studies suggest that some involved constructions of extensions of groups might lead to non wgsc groups. 73

4.3

Some examples of wgsc groups

In this section we study the class of wgsc groups. We prove that most geometric examples of groups are actually wgsc. It follows from [6, 7, 52] that: Corollary 4.3.1. 1. A group G is wgsc if and only if a finite index subgroup H of G is wgsc. 2. Let A and B be wgsc groups and C be a common finitely generated subgroup. The amalgamated free product G = A∗C B is wgsc. Moreover, if φ denotes an automorphism of a wgsc group A, then the HNN-extension G = A ∗C φ is wgsc. 3. All one-relator groups are wgsc. 4. Let G be a wgsc group. If G = A ∗ B then both A and B are wgsc. 5. Groups acting geometrically on a CAT(0) space are wgsc. 6. The groups from the class C+ (combable) in the sense of Alonso-Bridson ([1]) are wgsc. In particular automatic groups, small cancellation groups, semi-hyperbolic groups, groups acting properly co-compactly on Tits buildings of Euclidean type, Coxeter groups, fundamental groups of closed non-positively curved 3-manifolds are wgsc. Notice that all these groups have solvable word problem. 7. If a group has a tame 1-combing in the sense of [52] then it is wgsc. In particular, asynchronously automatic groups (see [52]) are wgsc. 8. If a group acts co-compactly on a polyhedron X, so that ske2 (X) is PL homeomorphic to the 2-skeleton of a wgsc space Y , then the group is wgsc. 9. Groups which are simply connected at infinity are wgsc ([6]). 10. Assume that 1 → A → G → B → 1 is a short exact sequence of infinite finitely presented groups. Then G is tame combable and thus wgsc ([7]). Remark 4.3.2. The last property above is an algebraic analog of the fact that the product of two open simply-connected manifolds is gsc. Moreover, if one of them is 1-ended then the product is simply connected at infinity.

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Proof. The proof can be obtained directly from the fact that wgsc and qsf are equals for groups. We give only some proofs which are easier with the wgsc condition. 1. Let X be the standard 2-complex associated to G. The subgroup H e in such a way that the finite complex X/H e acts on the universal cover X has e H as fundamental group. The result follows by observing that X and X/H have the same universal cover. 2. In order to prove the statement 2 we need another lemma. Lemma 4.3.3. Let M be a non-compact complex which is the union of two closed sub complexes K and L, such that K ∩ L is connected. If both K and L are wgsc, then M is wgsc. Proof. Let X be a compact subspace of M . Then X ∩(K ∩L) is compact and it is contained in a compact connected subspace Y ⊂ K ∩L. Let X 0 = X ∪Y , Xk = X 0 ∩K and Xl = X 0 ∪L. The pieces K and L are wgsc, then there exist BK ⊂ K and BL ⊂ L compact, simply connected sub complexes containing XK and XL respectively. Hence, the sub complex BK ∪ BL is a compact subset containing X which is simply connected by Van Kampen’s theorem. Then M is wgsc. We can now finish the proof of the statement 2. Take spaces KA and KB with fundamental groups A and B respectively. By attaching mapping cylinder from a bouquet of circles which represents the generators of C, one obtains a space with fundamental group G. The universal cover is a treelike fA and K fB joined along the C cover. If we consider arrangements of copies of K a finite number of pieces at time, the result follows from the preceding lemma, fA and K fB are wgsc. The case for since by hypothesis A and B and then K HNN extensions is similar. 3. The proof of the third assertion appeals to an interesting lemma on the structure of one-relator groups (see [46]). Lemma 4.3.4. Given any finitely generated one-relator group G there exists a finite sequence of finitely generated one-relator groups H1 , H2 , ..., Hn = G such that for each i < n, either Hi+1 or Hi+1 ∗ Z is an HNN extension of Hi over a finitely generated group, and such that H1 is either a free group or else it is isomorphic to a free product of a free group and a cyclic group. 75

This lemma reduces the proof to the case when a wgsc group splits into a (non-trivial) free product. 4. A standard argument using Grusko’s theorem shows that A and B are finitely presented. One can then choose standard 2-complexes PA and PB associated to them. The finite complex X obtained by a connection of PA e is and PB with an edge e has G as fundamental group. Its universal cover X fA and P fB joined by edges eg for any g ∈ G. It is an infinite union of copies of P fA (or P fB ) is wgsc to have the claim. Let sufficient to prove than one of these P e is wgsc one can find a subcomplex V be a compact subcomplex of it. Since X O of it containing V and simply connected. Actually its fundamental group fA with something is a free product of the fundamental group of O0 = O ∩ P 0 else. Since it is simply connected then O is also simply connected. Moreover fA . Thus A is wgsc. O0 contains V and it lies in P 5. One has just to observe that the metric balls in a CAT(0) space are simply connected since any loop in the ball can be contracted to the center of the ball by an homotopy inside the ball (by the convexity of the distance function), obtaining a wgsc space. In particular Gromov-hyperbolic groups are wgsc (in this case metric balls in the Rips complex are not only simplyconnected but even contractible). Proposition 4.3.5. A group which satisfies Cannon k-almost convex condition has Tucker property. Proof. Recall that a group G is k-almost convex if there exists an integer N = N (k) such that any two vertices x, y of the sphere of radius r in the Cayley graph of G with d(x, y) ≤ k, can be joined by a path of length ≤ N in the ball of radius r. Notice that a group is k-almost convex if and only if it is 2-almost convex. We want to show that almost-convex groups are Tucker. For convenience we suppose k is at least 3. Let B(r) be the ball of radius r centered at the identity of the Cayley graph X of G. We will prove that there exists a constant R = R(r) such that any loop out of B(r) is homotopically equivalent to a loop inside the subset B(R) − B(r). This shows that the fundamental group of X − B(r) is finitely generated (since the fundamental group of a finite complex is finitely generated). Recall that the isodiametric function of a group, following Gersten, is the infimal IG (q) so that loops of length q bound disks of diameter at most IG (q) in the Cayley complex. By hypothesis G is k-almost, thus we can define a constant I by setting I = IG (N (k) + 2). Set R = 2(r + N + I). 76

Let l be a edge loop based on B(R) − B(r), and suppose there is only one vertex v of maximal distance from the origin. Denote by M + 1 this distance, and suppose M + 1 > R. We want to homotope l to a loop in the ball of radius M . Observe that in our edge loop there are two vertices, a and b in the sphere S(M ), at distance one from v. Then d(a, b) ≤ 2, and so by hypothesis, there exists a path of length less that N contained in the ball of radius M joining them. Hence the loop consisting of this path followed by the edges (a, v) and (v, b), has length less than N + 2. By construction of I, any loop of length ≤ N + 2 bounds a disk within I from the loop. Hence the path (a, v, b) can be homotoped to a path inside the ball of radius M , by an homotopy of X − B(r). If the loop l has two adjacent vertices at maximal distance, then we have d(a, b) ≤ 3, and the same technique applies (since k is at least 3). Whenever our loop has three adjacent vertices, v, w and z at distance M + 1 from e, then one can do the following. Denote by a the vertex of the loop l setting in the sphere of radius M , adjacent to v, and b the vertex of l adjacent to z. If d(a, b) = 4, then one can find another point p (different from a and b) in the sphere of radius M at distance one from w. Then again d(a, p) ≤ 3 (resp. d(p, b) ≤ 3), and so by hypothesis one can find a path in the ball of radius M joining a with p (resp. p with b). Hence the path (a, v, w, z, b) can be pushed inside the ball B(M ) by an homotopy within X − B(r). Doing this inductively, one obtains that any loop l can be homotoped to a loop l0 in the ball of radius M , with an homotopy outside B(r). This procedure allows to contract any loop of X − B(r) to a loop inside B(R) − B(r), as wanted. This proposition allows us to give an alternative an simpler proof (bypassing the Dehn-type Lemma and the QSF-theory) of the main theorem of [60]: Corollary 4.3.6. If M is a closed, irreducible 3-manifold, with an infinite fundamental group G which is almost convex for some generating set C, then the universal covering space of M is homeomorphic to R3 . Proof. The previous proposition shows that the standard finite 2-complex X corresponding to the presentation C is Tucker, which means that for any ˜ π1 (X ˜ − D) is finitely generated. But this only depends compact D in X, ˜ has on the group G and not on the presentation chosen (by [52]), hence M Tucker property too. Moreover it follows from the result of Tucker ([72]) that ˜ is a missing boundary manifold. Finally, since it is a universal covering M ˜ is homeomorphic to of a closed 3-manifold, the results in [9] imply that M R3 . 77

Proposition 4.3.7. Tucker groups have small-depth (and so they are wgsc). Proof. Let G be a Tucker group and denote X its Cayley 2-complex. Let C be a compact subset of X. Since X is Tucker the fundamental group of the complementary of C if finitely generated, then one can find a finite compact subcomplex D containing all the generators of π1 (X − C). Let E be the compact subcomplex containing D and such that loops in D are trivial in E. Then the map π1 (D) → π1 (E) is zero, and, moreover, any loop in E − C is a loop in X − C and then homotopic (rel the base point within X − C) to a loop in D − C (since D contains the generators of π1 (X − C)). Hence G has small-depth. Proposition 4.3.8. Simply connected at infinity groups have small-depth. Proof. Let X be a finite complex with π1 (X) = G. Let C be a compact e By hypothesis, there exists D ⊇ C compact such that any subcomplex of X. e − D is trivial in X e − C. Choose a finite sub-complex E containing loop in X D and such that π1 (D) is zero in E. Any loop in E − C, based to a point in D − C, is a connected sum of the loops γ0 ⊂ (D − C) and γj ⊂ (E − D). Any loop in E − D in null-homotopic within X − C by hypothesis. Hence any loop in E − C is homotopic to a loop which lies in D − C. Thus G has small-depth.

4.3.1

Baumslag-Solitar groups

Baumslag-Solitar groups are given by the presentation B(m, n) =< a, b|abm a−1 = bn >, m, n ∈ Z It is known that B(1, n) are amenable, metabelian groups which are neither lattices in 1-connected solvable real Lie groups nor CAT(0) groups (i.e. acting freely cocompactly on a proper CAT(0) space). Nevertheless, B(1, n) is a HNN extension of Z by the injective endomorphism a → an and thus it is abelian-by-cyclic. Thus B(1, n) is wgsc. Notice that B(1, n) are not almost convex with respect to any generating set and not automatic either if n 6= ±1. Notice also that a group G which is simply connected at infinity should satisfy H 2 (G, ZG) = 0. Since this condition is not satisfied by B(1, n), for n > 1, these groups are not simply connected at infinity. The higher Baumslag-Solitar groups B(m, n) for m, n > 1 are known to be nonlinear, not residually finite, not virtually solvable, and, if (and only if) m and n are meshed, not Hopfian. Moreover, they are not automatic if 78

m 6= ±n, but they are asynchronously automatic. However, they are wgsc groups since they are extensions by a cyclic group and also 1-relator groups.

4.3.2

Solvable groups

If G is a finitely presented solvable group whose derived series is G . G(1) . G(2) . · · · . G(n) . G(n+1) = 1 If G(n) is finite the observe that G is wgsc if and only if G/G(n) is wgsc. Thus it suffices to consider the case when G(n) is infinite. According to Mihalik ([50]) if moreover G(n) has an element of infinite order then either G is simply connected at infinity or else there exists two groups Λ / G which is normal of finite index and F / Λ which is normal finite subgroup such that Λ/F is isomorphic to a Baumslag-Solitar group B(1, m). This will imply that Λ and thus G is wgsc. Thus, if G(n) has an element of infinite order then G is wgsc. But G(n) is finitely generated and abelian, then it has an element of infinite order unless it is finite. Hence solvable groups are wgsc.

4.3.3

Higman’s group

The first finitely presented acyclic group was introduced by G.Higman in 1951: H = hx, y, z, w|xw = x2 , y x = y 2 , z y = z 2 , wz = w2 i, where ab = bab−1 It follows that H is an iterated amalgamated product H = Hx,y,z ∗Fx,z Hz,w,x , Hx,y,z = Hx,y ∗Fy Hy,z where Hx,y = hx, y, y x = y 2 i is the Baumslag-Solitar group BS(1, 2) in the generators x, y. Here Fy , Fx,z are the free groups in the respective generators. The morphisms Fy → Hx,y , Fx,z → Hx,y,z and their alike are tautological i.e. send each left hand generator into the generator denoted by the same letter on the right hand. Remark that these homomorphisms are injective. The theorem above implies that H is wgsc. Observe that G is not sci [51].

4.3.4

Further examples

The Thompson groups. The first examples of infinite finitely presented simple groups were provided by R.Thompson in the sixties. We refer to [12] 79

for a thorough introduction to the groups usually denoted F , T and V . These are by now standard test groups. According to ([10, 27]) F is a finitely presented group which is an ascending HNN extension of itself. A result of Mihalik ([49], Th.3.1) implies that F is simply connected at infinity and thus wgsc. Remark 4.3.9. Notice that F is a non-trivial extension of its abelianization Z2 by its commutator [F, F ], which is a simple group. However [F, F ] is not finitely presented, although it is still a diagram group, but one associated to an infinite semi-group presentation. Grigorchuk groups. Grigorchuk defined in the eighties a group G that is finitely generated infinite torsion group of intermediate growth having solvable word problem (see [36]). This group is not finitely presented but Lys¨enok obtained ([47]) a nice recursive presentation of G as follows: G = ha, c, d | σ n (a2 ), σ n ((ad)4 ), σ n ((adacac)4 ), n ≥ 0i where σ : {a, c, d}∗ → {a, c, d}∗ is the substitution that transforms words according to the rules σ(a) = aca, σ(c) = cd, σ(d) = c It is therefore simple to find a finitely presented group G that contains G (see [36]) by using a HNN extension, which will be amenable but not elementary amenable. The endomorphism of G defining the HNN extension is induced by the substitution σ and thus the new group has the following finite presentation: G = ha, c, d, t | a2 = c2 = d2 = (ad)4 = (adacac)4 = 1, at = aca, ct = dc, dt = ci Conjecture: The group G is wgsc.

4.3.5

From discrete groups to aspherical manifolds

Proposition 4.3.10. Assume that there exists a finitely presented group Γ which is not wgsc. Then there exists a closed manifold M m of dimension m is not wgsc. If the group Γ were finite g m ≥ 4 whose universal covering M dimensional of type F∞ (i.e. with a finite dimensional classifying space) then the closed manifold M m could be chosen aspherical.

80

Proof. Let X be a compact manifold with boundary whose fundamental group is Γ (i.e. coming from thickening some 2-complex). We use Davis reflection group trick (see the appendix A below and [19]) in order to obtain e a closed manifold M m which retracts onto X. Moreover one knows that X fm properly retracts onto X. e Thus the results from ([30], is not wgsc and M m section 6) show that M cannot be gsc. Further if Γ were of type F∞ then the manifold M m would be aspherical. Remark 4.3.11. Notice that there exist infinite dimensional groups which are F∞ i.e. they have an infinite dimensional classifying space having only finitely many cells in each dimension. The first such example is Thompson’s group F that we encountered above (see [10]).

4.3.6

Balls and spheres in Cayley graphs

We wish to consider metric complexes whose balls or spheres satisfy a property which seems only slightly stronger than the tameness conditions above. Definition 4.3.12. The metric complex has π1 -small balls (resp. spheres) if there exists a constant C so that π1 (B(r)) (resp. π1 (S(r))) is normally generated by loops with length smaller than C, where S(r) denotes the sphere of radius r. We actually show that this put strong constraints on the group: Proposition 4.3.13. If a Cayley graph of a finitely presented group has π1 -small balls or spheres then the group has linear wgsc. Proof. A loop l in B(r) is null-homotopic in the Cayley graph. Thus the loop is freely homotopic to a loop l0 lying on the sphere S(r). For instance consider a thickening of the ball B(r) which is a manifold and then project down the intersection of the homotopy disk with the boundary. If we have π1 -small spheres then we can assume that the loop l0 is made of uniformly small loops on S(r) connected by means of arcs. A loop of length C in the Cayley graph bounds an disk of diameter I(C), where I is the isodiametric function of the group. The important think is that this is uniformly bounded. The loop l0 bounds therefore a disk which is disjoint from B(r − I(C)) and lies within B(r + I(C)). Do this for a system of loops lj which generate π1 (B(r)). Define A to be B(r) union a number of 2-disks attached along the loops lj . We define the map A → B(r + I(C)) by sending any disk into the nullhomotopy taking 81

first lj into lj0 and further null-homotopying the later. Then π1 (A) = 0 and the map is a π1 -resolution of B(r − I(C)). The same proof works for π1 -small balls. Corollary 4.3.14. A group having a Cayley graph with π1 -small balls or spheres has linear connectivity radius and solvable word problem. Proof. The linear connectivity radius is at most linear. Thus a loop in B(r) is null-homotopic in the Cayley graph only if it is so within B(cr). This can be checked by a finite algorithm. Remark 4.3.15. It is likely that hyperbolic groups have π1 -small balls and spheres. Remark 4.3.16. Recall that the isodiametric function of a group, following Gersten, is the infimal I(k) so that loops of length k bound disks of diameter at most I(k) in the Cayley graph. The rough equivalence class of I is a quasi-isometry invariant of the group. One could reformulate the definition of π1 -small spheres, in the case of a Cayley graph of a group, as follows: the group π1 (B(r)) is normally generated by loops of length ρ(r) where lim r − I(ρ(r)) = ∞

r→∞

Note that the limit should be infinite for any choice of I within its rough equivalence class. Then again groups verifying this condition are wgsc. However, it’s worthy to notice that I(r) should be non-recursive for groups with nonsolvable word problem, so that ρ(r) grows extremely slow if nonconstant. Moreover, if we only ask that the function r − I(ρ(r)) be recursive then the group under consideration should have again solvable word problem. In fact, we have by the arguments above the inequalities I(r − I(ρ(r)) ≤ f (r − I(ρ(r)) ≤ r + I(ρ(r)) < 2r and thus I(r) is recursive since bounded by the inverse of a recursive function.

4.3.7

Rewriting systems

Groups admitting a rewriting system form a particular class among groups with solvable word problem (see [40] for an extensive discussion). A rewriting system consists of several replacement rules wj+ → wj− 82

between words in a given alphabet. A reduction of the word w consists of a replacement of some subword of w according to one of the replacement rules above. The word is said irreducible if no reduction could be applied anymore. The rewriting system is complete if for any word in the generators the reduction process terminates in finitely many steps and is confluent which means that the irreducible words obtained at the end of the reduction is uniquely defined by the class of the initial word as an element of the group. Thus the irreducible elements are the normal forms for the group elements. If the rules are not length increasing then one calls it a geodesic rewriting system. We will suppose that the rewriting system consists of finitely many rules. Proposition 4.3.17. A finitely presented group admitting a complete geodesic rewriting system is wgsc. Proof. In [40] is proved that such a group is almost convex and thus wgsc (by proposition 4.6.3). Here is a shorter direct proof. We prove that actually the balls B(r) in the Cayley graph are simply connected. Observe first that in any Cayley graph we have: Lemma 4.3.18. The fundamental group π1 (B(r)) is generated by loops of length at most 2r + 1. Proof. Consider a loop ep1 p2 ...pk e based at the identity element e, sitting in B(r). Here pj are the consecutive vertices of the loop. There exists a geodesic γj that joins pj to e. It follows that the initial loop is the product of loops γj−1 pj pj+1 γj . Since pj ∈ B(r) all these loops have length at most 2r + 1. Consider now the Cayley graph of a group presentation that includes all rules from the rewriting system. This means that there is a relation associated to each rule wj+ → wj− . We claim that the balls B(r) are simply connected. By the previous lemma it suffices to prove that loops of length at most 2r + 1 within B(r) are nullhomotopic in B(r). Choose such a loop in B(r) which is represented by the word w in the generators. Since the loop is null-homotopic in the Cayley graph the word w should reduce to identity by the rewriting system. Let then consider some reduction sequence: w → w1 → w2 → · · · wN → e 83

Each word wi represents a loop in the Cayley graph. The step wj → wj+1 is a homotopy in which the first loop is slided across a 2-cell associated to a relation from the rewriting system. Further the lengths of these loops verify |wj | ≥ |wj+1 | since the length of each reduction is nonincreasing, by assumption. Thus |wj | ≤ 2r + 1 and this implies that the loop is contained within B(r). This proves that the reduction sequence above is a nullhomotopy of the loop w within B(r). Remark 4.3.19. The Baumslag-Solitar groups B(1, n) and the solvgroups (i.e. lattices in the group SOL) admit rewriting system but not geodesic ones ([40]), since they are not almost convex. Remark 4.3.20. More generally one proved in [40] that groups admitting a rewriting system are tame 1-combable (i.e. Tucker) and thus wgsc by [52]. Conjecture 4.3.21. Finitely presented groups that have solvable word problem are wgsc. Remark 4.3.22. An algorithm that solves the word problem yields also a specific nullhomotopy for a loop in the Cayley complex, but this nullhomotopy might be very distorted and there is no control of it if there are no additional assumptions.

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Chapter 5 The Tucker property 5.1

Introduction

The startpoint of our result is the series of papers by Poenaru ([58, 59, 60]), where the author, in his approach to the Covering conjecture, introduced the idea of killing 1-handles stably (i.e. by taking product with n-balls) to prove the simple connectivity at infinity of some open, simply connected 3manifolds. One of the ingredients used there is the Φ/Ψ-theory (introduced in [58]) of the equivalence relations forced by singularities of a non-degenerate simplicial map. Our aim is to use these techniques for non-simply-connected manifolds. Poenaru in [59] showed that if the product of an open, simply-connected 3manifold with the closed n-ball W 3 × Dn has no 1-handles, then the manifold W 3 is simply connected at infinity. Poenaru’s result was further extended in [29] (see also [32]) where it is proved that an open, simply-connected 3-manifold having the same proper homotopy type of a wgsc polyhedron is simply connected at infinity. These results are purely 3-dimensional and they cannot be extended to higher dimension, as stated. In fact, there exist compact, contractible n-manifolds M n for any n ≥ 4, such that Int(M n ) × [0, 1] is wgsc but Int(M n ) is not simply connected at infinity. Nevertheless, in [30], the above result was extended regardless to the dimension. In particular, it is proved that a non-compact manifold of dimension n, with n 6= 4, having the same proper homotopy type of a wgsc polyhedron is wgsc. If one wants to extend the wgsc property to non-simply connected spaces, one has to recall a concept developed by Tucker in his work around the 85

missing boundary problem for 3-manifolds ([72]). Definition 5.1.1. A polyhedron M is Tucker (or it has the Tucker property) if for every compact subpolyhedron K ⊂ M , each component of M − K has finitely generated fundamental group. Remark 5.1.2. A major difference between the wgsc condition and the Tucker property is that the former means that there is no 1-handles in some handlebody decomposition, while the second only tells us that some handlebody decomposition needs only finitely many 1-handles, but without any control on their number. It was shown in [72] that if a non-compact 3-manifold M is Tucker, then it is a missing boundary manifold, i.e. there exists a compact manifold N and a subset C of the boundary of N such that N − C is homeomorphic to M. The Tucker condition was also exploited by Mihalik and Tschantz in [52] where the authors introduced tame combings for finitely presented groups. It is shown that a group Γ is tame combable if and only if the universal covering e of some (equivalently, any) compact polyhedron X, with π1 X = Γ, is X Tucker. This is independent on the space X and hence the Tucker property can be seen as a group-theoretical notion. Moreover, Brick has shown that it is a geometric property for groups, in the sense that it is preserved by quasi isometries ([5]). Finally, we had the impression that the same techniques of [59] still work for non-simply connected 3-manifolds. In particular it might be true that if the product W 3 ×Dn of a (not necessarily simply connected) open 3-manifold W 3 with the closed n-ball has finitely many 1-handles, then W 3 is Tucker. Actually, we were able to prove the proper homotopy invariance of Tucker property without any restriction on the dimension (see [56]). Our main result is the following: Theorem 5.1.3. Let W n be a manifold of dimension n. If W n is proper homotopy equivalent to a Tucker polyhedron X k , then W n is Tucker. Remark 5.1.4. • This theorem implies a Poenaru-type result. Indeed, if W 3 × Dn has finitely many 1-handles then it is Tucker and hence, the theorem implies that W 3 is Tucker.

86

• Notice that the result of [52] implies the homotopy invariance of the Tucker property only for universal coverings. • We shall actually prove a stronger claim, namely that if W is proper homotopically dominated by a Tucker polyhedron then W is Tucker (see next section). • With respect to results of [59, 29] our result is somewhat soft because it does not use a Dehn-type lemma ([59]) specific to dimension three, and for that reason it holds true in any dimension. Corollary 5.1.5. Let V 3 be an open 3-manifold with H1 (V 3 , Z) = 0. Then V 3 has the proper homotopy type of a Tucker polyhedron if and only if V 3 is simple ended. Proof. The main theorem implies that V 3 is Tucker. Since V 3 is open, by [72] it follows that V 3 = int(M 3 ), where M 3 is a compact 3-manifold with boundary. Thus H1 (M 3 ) = H1 (V 3 ) = 0 and then the boundary of M 3 is made of spheres.

5.2

Preliminaries on Poenaru’s Φ/Ψ-theory

This section is devoted to recall some of the basic facts on the Φ/Ψ-theory introduced by Poenaru in [58] as a useful tool in his approach to the covering conjecture (see [59, 60]). The object is to investigate, in the context of a non-degenerate simplicial map f : X → M from a not necessarily locally finite simplicial complex X to a manifold M , an equivalence relation on X which is the smallest possible such relation, compatible with f and killing all the singularities of f . The simplicial complex X will be endowed with the weak topology (i.e. a subset C is closed if and only if C ∩ {simplex} is closed), and this makes the map f continuous. Let X be a simplicial complex of dimension at most n, not necessarily locally finite, but with countably many simplexes. Let M be a n-manifold and f : X → M be a non-degenerate simplicial map. Definition 5.2.1. A point x ∈ X is not a singular point if f is an embedding in a neighborhood of x. Write Sing(f ) for the set of singular points of X. Definition 5.2.2. The equivalence relation Φ(f ) ⊂ X × X defined by f is given by (x, y) ∈ Φ(f ) ⇔ f (x) = f (y). 87

Note that we are interested only in equivalence relations which are such that, if x, y are two points belonging to two simplexes of X of the same dimension, with f (x) = f (y), then (x, y) ∈ R implies R identifies the two simplexes. Such equivalence relations have the property that X/R is a simplicial complex; the induced map X/R → M is also simplicial. Poenaru has shown that, starting with Φ(f ), one can construct, by folding maps, “the smallest” equivalence relation Ψ(f ) ⊂ Φ(f ) killing all the singularities without changing the topology of X. More precisely: Proposition 5.2.3. There exists a unique equivalence relation Ψ ⊂ Φ such that: • if f denotes the induced map f : X/Ψ(f ) → M , then f has no singularities (i.e. f is an immersion). • there exists no other equivalence relation Ψ1 ⊂ Ψ having the same properties as Ψ and such that Ψ1 (f ) $ Ψ(f ). Hence Ψ(f ) is the smallest equivalence relation compatible with f which kills all the singularities. Moreover, while the passage from X to X/Φ (which, by definition, is just f (X)) destroys all the topological information, this does not happen for Ψ, in fact: Proposition 5.2.4. The projection map π : X → X/Ψ(f ) is simplicial and it induces a surjection on fundamental groups π∗ : π1 (X) → π1 (X/Ψ(f )). Now, we give an idea how to obtain such a Ψ. Let f : X → M be a non-degenerate simplicial map. If Sing(f ) = ∅ then Ψ is trivial (Ψ = Diag(X, X)); if not, there exist a point x ∈ Sing(f ) and two distinct simplexes of the same dimension σ1 and σ2 such that x ∈ σ1 ∩ σ2 and f (σ1 ) = f (σ2 ). In this case we consider the quotient relation ρ1 obtained by identifying σ1 and σ2 (ρ1 is called a folding map). Now, if the induced map f1 : X/ρ1 → M (that is non-degenerate and simplicial too) has no singularities, then Ψ(f ) = ρ1 . Otherwise, we proceed as above. If our X is finite, ∞ the process stops after finitely many steps, and we obtain Ψ(f ) = ∪ ρn . n=1 In the infinite case, we get a sequence of quotients corresponding to an increasing sequence of equivalence relations: ρ1 ⊂ ρ2 ⊂ · · · ⊂ ρn · · · ⊂ Φ(f ). The union ρω = ∪ρi is a subcomplex and hence it is closed in the weak topology. The quotient Xω = X/ρω has a simplicial structure, and the 88

natural map fω : Xω → M is simplicial. We can now pick up a point in Sing(fω ) and go on with a transfinite sequence which stops at the first ordinal where there are no more singularities left: ρ1 ⊂ ρ2 ⊂ · · · ⊂ ρn · · · ρω ⊂ ρω+1 ⊂ · · · Φ(f ). It follows from the previous proposition that Ψ(f ) = ∪i ρi . This also shows that ∪i ρi is well defined, in the sense that it is independent of the choices made in the construction. Remark 5.2.5. The intermediary spaces Xi are actually simplicial complexes of a more general type: these are multicomplexes in the sense of Gromov ([58]). Consider now an infinite but countable X. The next lemma gives us a really manageable version of Ψ(f ) since it says that one can always choose the sequence of foldings so that ρω = Ψ(f ), i.e. without using ordinal numbers. Proposition 5.2.6. Let X be an at most countable complex. Then there exists a good manner to proceed with the folding maps in order to have ∞

Ψ(f ) = ∪ ρn . n=1

Rigorous definition of Ψ(f ). Define M 2 (f ) ⊂ Φ(f ) to be the set of double points of f , i.e. all the (x, y) ∈ (X × X) − Diag X such that c(f ) = M 2 (f ) ∪ Diag(Sing(f)) ⊂ Φ(f). Clearly, f (x) = f (y), and denote M c(f ) has a natural structure of simplicial complex. We will endow M c(f ) M with the weak topology. b be a subset of M c. We say that R b is admissible if the subset R = Let R b R∪Diag X is an equivalence relation satisfying the following: if f (x) = f (y) with x ∈ σ1 , y ∈ σ2 and (x, y) ∈ R, then R identifies the simplexes σ1 and b is a simplicial complex. σ2 . Note that this means that X/R c by We can now define a new topology (called the Z-Topology) on the set M deciding that the Z-closed subsets are the finite unions of admissible subsets. Finally one can prove the following equality: the equivalence relation Ψ is the smallest admissible equivalence relation containing Diag(Sing(f)), i.e. Ψ(f ) is the Z-closure of Diag(Sing(f)).

5.3 5.3.1

The proof Outline of the proof

We first recall the following definitions from [29]: 89

Definition 5.3.1. A polyhedron M is (proper) homotopically dominated by the polyhedron X if there exists a P L map f : M → X such that the mapping cylinder Zf = M × [0, 1] ∪ X (properly) retracts on M . f

Remark 5.3.2. Observe that a proper homotopy equivalence is the simplest example of proper homotopy domination. Definition 5.3.3. An enlargement of the polyhedron E is a polyhedron X which retracts properly on E, i.e. such that i E ,→ X id& .π E where i is a proper P L embedding and π is a proper P L map. The main ingredient of the proof of the main theorem is the following lemma. Lemma 5.3.4. It suffices to prove the theorem for the case when X k is an enlargement of W n . Let us introduce some more terminology. Definition 5.3.5. A polyhedron P has a strongly connected n-skeleton if any two n-simplexes of P can be joined by a sequence of n-simplexes such that consecutive ones have a common (n − 1)-dimensional face. The polyhedron is n-pure if its n-skeleton is the union of its n-simplexes. Finally, P is called n-full if it is both n-pure and has a strongly connected n-skeleton. We now turn to the second reduction. Lemma 5.3.6. We can assume that the polyhedron X k is n-full. Now, the hypothesis of the lemma 5.3.4 provides us with a proper P L embedding W n ,→ X k and a proper surjection π : X k → W n such that π ◦ i = id. Furthermore we can suppose that X k is n-full thanks to the lemma 5.3.6. Lemma 5.3.7. There exist triangulations τW of W n and respectively θX of the n-skeleton of X k and a map λ : θX → τW such that: • λ is proper, simplicial, generic and non-degenerate (i.e. its restriction to any simplex is one-to-one) 90

• λ ◦ i = id • θX is Tucker. One derives that θX is an enlargement of τW , when the natural projection map is replaced by λ. Now we use the Φ/Ψ-theory introduced by Poenaru in [58]. Denote by λ : θX /Ψ → τW the simplicial map induced by λ. Lemma 5.3.8. The equality Φ(λ) = Ψ(λ) holds. Remark 5.3.9. Roughly speaking, this equality means that it is possible to exhaust all singularities of λ(θX ) by a countable union of folding maps.

5.3.2

Proof of the theorem using the lemmas

To simplify the notation, we denote by λC the restriction of λ to some subset C, and by ΨC or ΦC the equivalence relations of λC . Fix a connected compact subset K of W . By compactness arguments, one can find another compact subset L of θX such that λL (L) ⊃ K, and a third compact subset P ⊂ θX such that λ−1 (λL (L)) ⊂ P . This implies that if (x, y) ∈ M 2 (λ) with x ∈ i(K), then y ∈ P . Hence i(K) ⊂ P/ΦP . The last lemma (5.3.8) says that the equivalence relation Φ(λ) can be obtained by a countable union of folding maps. This implies that any compact subset of X involves only a finite number of these foldings. Hence, for any compact subset P of X there exists a big compact subset Pe such that ΨPe = ΦP . Furthermore we have the following diagram of maps λPe i(K) ⊂ P/ΦP = P/ΨPe ⊂ Pe/ΨPe → τW

Since the map λPe is an immersion and no double point of it can involve P (thanks to the equality ΦP = ΨPe ), we have i(K) ∩ M2 (λPe ) = ∅ and then λ i(K) ⊂ Pe/ΨPe ⊂ θX /ΨPe −→ τW .

Since θX is Tucker (lemma 5.3.7), the fundamental group π1 (θX − i(K)) is finitely generated and therefore the proposition 5.2.4 stated before implies 91

that π1 (θX − i(K))/Ψ(λ) is also finitely generated. Now, since i(K) ∩ M2 (λ) = ∅, one obtains that (θX − i(K))/Ψ is exactly θX /Ψ − i(K). From the equality of the lemma 5.3.8, we derive that θX /Ψ = θX /Φ. By definition of Φ, θX /Φ is just the image of θX by λ, namely τW . It follows that the fundamental group π1 (τW −K) is isomorphic to π1 (θX − i(K)), and hence finitely generated. The Tucker condition for W is then verified.

5.4 5.4.1

Proof of the lemmas Proof of the lemma 5.3.4

Let f : W → X be a proper homotopy equivalence. Then the mapping cylinder Zf = W ×[0, 1]∪X has a strong deformation retraction on W . Notice f

that X is also a strong deformation retract of Zf , by using the following retraction r(x, t) = f (x) for (x, t) ∈ W × [0, 1] and r(y) = y for y ∈ X. In particular r is a homotopy equivalence. Lemma 5.4.1. If X is Tucker then Zf is Tucker. Proof. Let C be a compact subset of Zf , and write K = r(C) ⊂ X. The compact subset r−1 (K) of Zf is the mapping cylinder of f |f −1 (K) : f −1 (K) → K, and it strongly retracts on K. Then the fundamental group π1 (Zf − r−1 (K)) is isomorphic to π1 (X −K) which is finitely generated by hypothesis. This implies that π1 (Zf − C) is also finitely generated as claimed. It follows that Zf is an enlargement of W , and the proof of the first lemma is achieved.

5.4.2

Proof of the lemma 5.3.6

We observe that if X is an enlargement of W , then X × Dp is also an enlargement of W for any p. The lemma follows now directly from the more precise statement below (see [29]): Lemma 5.4.2. If X is path-connected, then X × Dp is p-full. Proof. One has that skek (X n × Dk ) = skek (skek (X) × Dk ) because higher dimensional simplexes of X n do not contribute to the k-skeleton of X n × Dk . First there are no isolated 0-simplexes since X n is connected. Notice that 92

we have to choose a triangulation of skek (X n ) × Dk ) which is compatible with the product PL structure. Such a triangulation is highly not unique and the homeomorphism type of skek (skek (X n ) × Dk ) depends on our choice. However the property of being k-full depends only on the PL structure (since it is invariant at subdivisions). The product skek (X n ) × Dk is naturally divided into product cells of the form σ1r × σ2s , where σ1r , σ2s are simplexes (of dimensions r and s respectively) of skek (X n ) and Dk respectively. One identifies here Dk with one k-simplex. A triangulation of the product will consists of a triangulation of each product cell, with obvious compatibilities on the boundaries. Notice that the product cell complex admits a simplicial subdivision with no extra vertices (see Lemma 1.4, p.11 from [41]). We will consider subdivisions of a such triangulation below. Assume that we have a m-simplex σm ⊂ σ1r × σ2s of skek (X n ) × Dk , with 1 ≤ m ≤ k. The simplex σ2s is contained in some k-simplex σ2k . Since the product cell σ1p × σ2k is a PL ball of dimension at least k + 1 the star S(σ m ) of σ m contains a PL ball of dimension at least k + 1, and so σ m is included in some k-simplex of the triangulation. This shows that X n ×Dk is k-pure. Consider now the following equivalence relation on the points of a polyhedron: two points are equivalent (we should write k-equivalent to be more precise, but we drop the reference to the dimension) if they belong to some closed k-simplexes which can be joined by a chain in which consecutive k-simplexes have a common face. Equivalence classes of points form the thick connected components of the polyhedron. Due to the invariance at subdivision the thick components depend only on the adjacent PL structure, and not on the particular triangulation. Further, as we observed previously, any k-simplex σ k of a triangulation of X n × Dk is contained in a product cell σ1p × σ2k , which is a PL ball of dimension at least k + 1. Then all its points are k-equivalent. In particular the points of σ k are equivalent to those belonging to some particular k-simplex, for instance the (vertical) k-simplex σ1o × σ2k , where σ1o is a vertex of σ1p . Any two vertices of X n can be joined by a path, and we can construct a tube having the shape {path} × σ2k connecting the vertical k-simplexes lying over the endpoints. Using this tube (suitably triangulated) we obtain a chain connecting the two vertical k-simplexes. This shows that two points on arbitrary k-simplexes are equivalent, hence the k-skeleton is strongly connected.

5.4.3

Proof of the lemma 5.3.7

This lemma is a consequence of some general results of approximation of P L maps by non-degenerate maps. In fact the lemma 4.4 from [41] and the 93

remark which follows it state that: Lemma 5.4.3. Let f : P → Q be a P L map, Q a P L manifold and P a P L space with dimP ≤ dimQ. Let P0 ⊂ P be a closed subspace. Suppose that f |P0 is non-degenerate. Then f is homotopic to f 0 rel P0 , where f 0 is a nondegenerate P L map and f 0 (P − P0 ) ⊂int(Q). Moreover, given : P → R+ a positive continuous function, we may insist that ρ(f (x), f 0 (x)) < (x), for all x, where ρ is a given metric for the topology of Q, and the homotopy between f and f 0 does not move points farther then a distance apart at any moment. Furthermore, if f is proper we can ask that f 0 be proper, too. An application of this lemma when P is the n-skeleton of X, P0 = W ⊂ P and Q = W and f = π|P , gives a map λ = π 0 which is non-degenerate and generic. Moreover, the theorem 3.6 from [41] says that one can subdivide the n-skeleton of X and τW to make λ simplicial. Now, one has to observe that the Tucker property is preserved by subdivisions (since they do not affect the fundamental group) and taking the k-skeleton with k > 1 (indeed, if C is a compact subset of the k-skeleton, Xk of X, then the k-skeleton of X − C is Xk − C and so π1 ((X − C)k ) = π1 (Xk − C) = π1 (X − C) which is finitely generated if X is Tucker). This proves that the n-skeleton of X, θX , is Tucker.

5.4.4

Proof of the lemma 5.3.8 λ

By definition of Ψ, the map: θX /Ψ −→ τW is an immersion. We claim that it is a simplicial isomorphism. Consider the commutative diagram below θX /Ψ

λ

−→

τW

i - % id τW /Ψ If we prove that i is onto, then automatically λ is injective. If the contrary holds, we could find a n-simplex σ of θX /Ψ such that Int(σ) ∩ Im(i) = ∅, because X, and hence θX , is n-pure. Moreover, if σ1 and σ2 are two simplexes of θX /Ψ, arbitrary lifts of them to θX are connected by a chain of n-simplexes (since θX is n-full). The projections of the intermediary simplexes form a chain in θX /Ψ because the projection map is simplicial (Ψ is a composition of folding maps) and non-degenerate (as λ). Thus there exists some n-simplex σ such that Int(σ) ∩ Im(i) = ∅ = 6 σ ∩ Im(i) 94

Moreover λ(i(τW /Ψ)) = τW , and then any point in the boundary ∂σ ∩ Im(i) would be a singular point of λ, which is not possible since λ is an immersion. Now we want to show that Ψ = Φ. We have two bijections θX /Ψ → τW (already proved), and θX /Φ → τW (by definition of Φ), and an inclusion Ψ ⊂ Φ, which induce a bijective map θX /Ψ → θX /Φ. Hence the equality Φ = Ψ holds.

5.5

Proper homotopy invariance of the qsf

The technique above works for the qsf property, in fact: Theorem 5.5.1. A non-compact manifold is qsf if and only if it is proper homotopically dominated by a qsf polyhedron. Proof. Let us admit the above lemmas hold true. Let K be a compact subset of W ; as before there exist K1 compact such that K ⊂ λK1 (K1 ) and K2 compact such that i(K) ⊂ λ−1 (λK1 (K1 )) ⊂ K2 . This implies that if (x, y) ∈ M 2 (λ) with x ∈ i(K) then y ∈ K2 . Thus K ⊂ K2 /ΦK2 . By a compactness argument, and since Φ(λ) = Ψ(λ), for any compact subset K2 of θX , there exists another compact K3 such that Φ(λK2 ) = Ψ(λK3 )/K2 . Hence we obtain i(K) ⊂ K2 /ΦK2 = K2 /ΨK3 ⊂ K3 /ΨK3 . This means that i(K) does not meet the double points of λK3 (i.e. the map λ restricted to K3 is an immersion over i(K)). Now, by hypothesis, θX is qsf. Consider then a resolution K3 for K3 , i.e. f : K3 → X with K3 compact and simply connected, and containing a homeomorphic image of K3 . We claim that this is also a resolution for K in W . In fact the map f 0 composed by f followed by the map λK3 , is a map from K 3 to W . Furthermore the compact K is contained in f 0 (K3 ) (since f (K3 ) contains K3 ). The crucial observation is that K is homeomorphic to λ−1 K3 K since K ∩ M2 (λK3 ) = ∅. Then the complex K3 contains a homeomorphic image of K since λ−1 K3 K is contained in K3 , and K3 is homeomorphic to f −1 K3 . This ends the proof. We are left with the proof of lemmas. All the lemmas are exactly the same, except for the lemma 5.3.4 (actually 5.4.1) for which we need a proof. Lemma 5.5.2. If X is qsf then Zf is qsf. Proof. If C is a compact subset of Zf , consider K = r−1 r(C) and a resolution for r(K). This means that there exists a compact simply-connected complex 95

H and a map h : H → X such that r(K) ⊂ h(H) and hh−1 r(K) : h−1 r(K) → r(K) is a homeomorphism. Let A = r(K) ∩ f (W ) and define a new complex H 0 as follows: H ∪h−1 A h−1 (A) × [0, 1]. This complex is compact and simply ˆ defined as h in H and which sends (h−1 (x), ε) to connected and the map h (x, ε), is a resolution for r(K). Remark 5.5.3. The same argument shows that the Dehn-exhaustibility is a proper homotopy invariant. Notice that if one replace the hypothesis X qsf by wgsc, the conclusion W wgsc does not follows. In fact, instead of a resolution of the compact K3 , one obtains that K3 is contained in K4 /ΨK4 with K4 simply connected. Even if K4 /ΨK4 is simply-connected, we cannot say that the image λK4 (K4 ) is still simply-connected!

96

Appendix A Compact aspherical manifolds not covered by Rn The aim of this section is to show the following remarkable theorem: Theorem A.0.4. For any n ≥ 4 there exists a closed aspherical n-manifold whose universal covering is not homeomorphic to the Euclidean space. The construction is due to M.Davis (see [18] and [19]). The plan is the following: define a process for converting a simplicial complex L (with possibly interesting topology) into a finite cubical complex XL and a group W acting nicely on its universal cover. Next, consider the case when L is the (nonsimply connected) boundary of a contractible manifold (a Poenaru-Mazur eL is contractible and non-simply connected at inmanifold), and show that X finity. In such a case, one has to modify XL in order to obtain a contractible eL (with a cocompact reflection group acting on it) which is not n-manifold X simply connected at infinity. Let us recall some definitions we will use in the sequel. Definition A.0.5. A cubical cell complex is a regular cell complex in which each cell is combinatorially isomorphic to a standard cube of some dimension. (Just like simplicial complexes, if two cubes intersect then they do so in a proper face). Let P be a cubical complex and v a vertex of P . The link of v in P can be identified as the space of all directions based on v, or equivalently, one can think of Lk(v, P ) as the metric unit sphere about v. Notice that a neighborhood of a vertex v is homeomorphic to the cone on the link of v, Lk(v, P ). 97

Definition A.0.6. A cell complex L is a flag complex if • each cell in L is a simplex • L is a simplicial complex (i.e., the intersection of two simplices is either empty or a common face of both) • the following condition holds: (∗) Given any subset {v0 , v1 , . . . vk } of distinct vertices in L such that any two are connected by an edge, then {v0 , v1 , . . . vk } spans a k-simplex in L. Remark A.0.7. Gromov uses the phrase L satisfies the “no ∆ condition” to mean that the simplicial complex L satisfies (∗). Other authors said L is “determined by its 1-skeleton” to mean that (∗) held. One of the ingredient in the proof will be the next lemma: Lemma A.0.8 (Gromov’s Lemma). A cubical complex is non-positively curved if and only if the link of its vertices is a flag complex. Notice that such a cubical complex is aspherical, since the universal covering of a non-positively curved space is CAT(0), and hence contractible. Definition A.0.9. A Poenaru-Mazur manifold is a compact, contractible nmanifold with non-simply connected boundary. These manifolds exist for any n ≥ 4 ([57]). Definition A.0.10. A homology n-sphere is a manifold having the same homology as the sphere S n .

A.0.1

A cubical complex with prescribed links

The point of this subsection is to describe a simple construction which proves the theorem below. Theorem A.0.11. Let L be a finite simplicial complex. Then there is a finite piecewise-Euclidean cubical complex XL such that the link of each vertex of XL is isomorphic to L.

98

Proof. Notation. For any set I, let RI denote the Euclidean space consisting of all functions f : I → R. Let I = [−1, 1]I denote the standard cube in RI . Suppose that I = V ert(L), the vertex set of L. If σ is a simplex of L, let I(σ) denote the vertex set of σ. Let RI(σ) be the linear subspace of RI consisting of all functions f : I → R with support in I(σ). The Construction. The space XL is the subcomplex of I consisting I(σ) of all faces of I which σ in L. In S are parallel to R I for some simplex other words, XL = σ∈Simp(L) { faces of parallel to RI(σ) }. Note that Vert(XL ) = {+1, −1}I and that if v ∈ Vert(XL ), then the link Lk(v; XL ) w L. This proves the theorem. Lemma A.0.12. If L is homeomorphic to the sphere S n−1 then XL is a n-manifold. Proof. A neighborhood of a vertex in a cubical complex is homeomorphic to the cone on its link. The link of each vertex of XL is homeomorphic to a sphere. Since the cone on S n−1 is a disk, then any point of XL has a neighborhood homeomorphic to Rn . Let (ei )i∈I denote the standard basis of RI . For each i ∈ I, let ri : RI → RI be the linear reflection which sends ei to −ei and ej to ej . The group G of linear transformation generated by (ri )i∈I is isomorphic to (Z/2)I . The cube I is stable under G. Define K = XL ∩ [0, 1]I , and put Ki = K ∩ {xi = 0}, that is to say, Ki is the intersection of K with the hyperplane fixed by ri . For each i ∈ I, let si be a symbol and consider the Coxeter group W generated by S = (si )i∈I with relations si smij = 1, where mij = 1 if i = j, 2 if {i, j} ∈ j Edge(L) and ∞ otherwise. The pair (W, S) is called a right-angled Coxeter system. For any subset J ⊂ I, let WJ be the subgroup generated by {(si )i∈J }. For any point k ∈ K, let I(k) = {i ∈ I|k ∈ Ki } and put Wk = WI(k) . Give to W the discrete topology and define the space U (W ; K) to be the quotient space of W × K by an equivalence relation v, where two pairs (w; k) ∼ (w0 ; k 0 ) are equivalents if and only if k = k 0 and w−1 w0 ∈ Wk . Thus the space U (W, K) is the space formed by gluing together copies of K one for each element of W , the copies w × K and wsi × K being glued along the subspaces w × Ki and wsi × Ki . The equivalence class of (w; k) in U (W ; K) is denoted by [w; k]. The group W acts naturally on U (W ; K). Let φ : W → (Z/2)I be the natural homomorphism si 7→ ri , and let ΓL be its kernel. Lemma A.0.13. The space U (W ; K) is connected, simply connected and the map p : U (W ; K) → XL defined by [w, k] 7→ φ(w)k is a covering projection. 99

The proof is not difficult, actually it follows from the fact that K and Ki are contractible (they are cone on L). This means that the space U (W ; K) is the universal cover of XL . Hence, by using Gromov’s lemma, we have that: Theorem A.0.14. Suppose L is a flag complex. Then XL is a cubical complex of nonpositive curvature; its (contractible) universal cover is UL = U (W ; K) and its fundamental group is ΓL .

A.0.2

The example

Before construct Davis’ example, we need to recall some facts. First, for any n ≥ 3, there are homology n-spheres which are not simply-connected; moreover, see [26], every homology (n − 1)-sphere bounds a contractible topological n-manifold X (with boundary). Thus the problem of finding a closed aspherical manifold not covered by Rn is not the existence of fake contractible manifolds (they exist), but rather if such an example can admit a discrete cocompact group of transformations. Theorem A.0.15. Suppose L is a homology (n−1)-sphere, with L not simply connected. Then the space UL = U (W ; L) is not simply connected at infinity. Proof. The idea is to calculate π1 (UL − Cn ) where (Cn )n is an increasing family of compact subsets exhausting UL . Consider a triangulation of L as a flag complex. Then UL admits a CAT(0) metric, one can take Cn = B(xo , n), where x0 is a base point and B(x0 , n) is the ball of radius n about x0 . Davis proved that the sphere Sx0 (n) is homeomorphic to the connected sum of the link of all the vertices inside B(xo , n) − Sx0 (n). Hence Sx0 = n-connected sum of L. Now, since the geodesic contraction provides an homotopy equivalence between UL −B(xo , n) and the sphere Sx0 (n), one has that the fundamental group of the complementary of a n-ball is the free product of n copies of π1 (L) 6= 1. Then the fundamental group at infinity of UL cannot be trivial: it is an inverse limit of free products of more and more copies of π1 L (this is the first example of a group whose fundamental group at infinity is not either trivial, Z or an infinite rank free group). Remark A.0.16. The same can be done with a Poenaru-Mazur manifold X by considering a barycentric subdivision L of its boundary. Actually this not yet a counterexample of the covering conjecture, since the space XL is not necessary a manifold. But one can modify XL to have 100

a manifold. In fact, XL = U (W ; K)/ΓL , where K is homeomorphic to the cone of L, and L is a homology (n − 1)-sphere. As already noticed, any such a sphere bounds a contractible n-manifold K 0 with boundary ∂K 0 = L. Idea: hollow out each copy of K and replace it with a copy of K 0 . Since ∂K 0 = L we can define Ki0 = Ki ⊂ ∂K 0 . Then U (W ; K 0 ) is a manifold since there are no longer cone points. Furthermore the map f from K 0 to K which is the identity on ∂K 0 and a homotopy equivalence rel ∂K 0 , extends to a W equivariant proper homotopy equivalence from UL0 to UL (one sends a class [w, k] to [w, f (k)]). Since UL is contractible and non sci, it follows that UL0 is a contractible, non sci manifold (being sci is a proper homotopy invariant). Moreover XL0 = U (W ; K 0 )/ΓL is an aspherical manifold with universal cover UL0 . Hence we have: Theorem A.0.17. For each n ≥ 4 there is a closed aspherical n-manifold not covered by Rn .

A.0.3

The reflection trick

In this section we generalize the above manipulation of XL for any space X with boundary. Suppose X is a space and that ∂X is a subspace which is homeomorphic to a polyhedron. Let L be a triangulation of ∂X as a flag complex with vertex set I, and let XL be the cubical complex associated to L. For each i ∈ I, put Xi = Ki where Ki is the previously defined subcomplex of XL ∩ [0, 1]I . Let G, W and ΓL be the groups defined previously. Set UX = U (W ; X)/ ∼

and

MX = UX /ΓL

where the equivalence relation is defined exactly as before. We record that G w (Z/2)I acts on MX and that the orbit space is X, since MX can be constructed as (G × X)/ ∼. Let r : MX → X be the orbit map. Since X can also be regarded as a subspace of MX (namely by the map x → [1, x]), we have the following theorem. Theorem A.0.18. The map r : MX → X is a retraction. Corollary A.0.19. The fundamental group π1 (X) is a retract of π1 (MX ). Theorem A.0.20. If X is a compact n-dimensional manifold with boundary (= ∂X), then MX is a closed n-manifold.

101

Proof. For each x ∈ X, let σ(x) = {i ∈ I|x ∈ XI }. A neighborhood of x in σ(x) σ(x) ∂X has the form Rn−k ×R+ where k = Card(σ(x)), where R+ denotes the positive quadrant in Rσ(x) where all coordinates are nonnegative. It follows σ(x) that a neighborhood of (1, x) in MX has the form Rn−k × (Jσ(x) × R+ )/ ∼ which is homeomorphic to Rn . Thus, MX is an n-manifold. We also have that UX → MX is a regular covering with group of deck transformation ΓL . The proof of the next proposition is the same as before (see [19]). Proposition A.0.21. (i) If X is simply connected, then UX is the universal cover of MX and hence, π1 (MX ) = ΓL . (ii) If X is aspherical, then so is MX . The reflection trick and the construction of MX can be also used to show that: Proposition A.0.22. Given any recursively presented group H, there is a one-ended right-angled Coxeter group W such that G < π1∞ (W ). Here is another application. Let G be a group whose classifying space BG is a finite complex. Then we can thicken it to a compact manifold with boundary. This means that we can find a compact manifold with boundary X which is homotopy equivalent to BG. The proof goes as follows. First, up to homotopy, we can assume that BG is a finite simplicial complex. The next step is to piecewise linearly embed BG in some Euclidean space Rn . So, we can assume BG is a subcomplex of some triangulation of Rn . Finally, possibly after taking barycentric subdivisions, we can replace BG by a regular neighborhood X in Rn . The reflection group trick can then be summarized as follows. Start with a group G with a finite BG. After thickening we may assume that BG is a compact manifold with boundary X. Triangulate ∂X as a flag complex L. Then MX is a closed aspherical manifold which retracts onto BG. S. Weinberger has noted that there are examples of finitely presented groups G with unsolvable word problem such that BG is a finite 2-complex. Since any group which retracts onto such a group also has unsolvable word problem, the reflection group trick gives the following result. Corollary A.0.23. For each n ≥ 4, there is a closed aspherical n-manifold the fundamental group of which has unsolvable word problem.

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Appendix B Discrete subgroups of Lie groups There are two sharply distinct classes of closed subgroups of Lie groups. The first is where the subgroup H is connected; then H is a Lie subgroup determined by its Lie algebra. The opposite case is when a subgroup Γ is discrete in G: in such a case, the linear structure of L(G) cannot be used in the study of Γ as the infinitesimal information contained in L(G) is ≡ 0. The basic reference on this subject is the book of Raghunathan [64]. Consider for example G = R2 and Γ = {ax + by : a, b ∈ Z}, or Γ = {mx : m ∈ Z}; in the first case the quotient space is the torus, while the second is a non-compact cylinder. When G is the (Lie) group of isometries of a simply connected, Riemannian manifold X, then the structure of any discrete subgroups Γ of G is reflected in the geometry of V = X/Γ. If Γ is torsion-free then it acts freely on X and X is the universal covering of V .

B.0.4

Uniform lattices

Let G be a Lie group. A discrete subgroup Γ ⊂ G is co-compact if the quotient space G/Γ is compact (or equivalently, there exists a compact subset D ⊂ G whose Γ-translates cover G). One can also say that Γ is a net in G, i.e. there exists a constant c such that Γc = {g ∈ G : d(g, Γ) ≤ c} = G. Definition B.0.24. A lattice of a group G is a discrete subgroup Γ ⊂ G satisfying one of the following equivalent properties: (1) Γ\G holds a finite left G-invariant measure coming from the Haar measure of G. 103

(2) Γ\X has finite volume. Moreover, if Γ is co-compact, then Γ is called uniform lattice or cocompact lattice. Otherwise Γ is called non-uniform lattice. Observe that a discrete co-compact subgroup has finite volume, while the converse is not true in general. Actually, it holds for NIL, SOL Lie groups and for Rn , while for semi-simple Lie groups it fails. In fact, the subgroup Γ = SLn (Z) ⊂ SLn (R) has finite co-volume for n ≥ 2 but it is not cocompact. Theorem B.0.25. Let G be a connected real Lie group and let Γ be a lattice. Then Γ is finitely generated. This theorem has no simple proof (for non-uniform lattices). We would like just to remark that if one make the assumption that Γ is co-compact, then Γ is finitely presented, and quasi-isometric to G. Definition B.0.26. A lattice Γ is irreducible if it does not contain finite index subgroups which split as a lattice product. Definition B.0.27. The rank of a Lie group G is the maximal dimension of a flat, that is a totally geodesic subspace isometric to an Euclidean space. Notice that a symmetric space has rank one if and only if its sectional curvature is strictly negative.

B.0.5

Non-uniform lattices and Q-rank

An important notion in understanding large-scale geometry of lattices (in particular non-uniform lattices) is that of horoball. One can define this notion in much more general spaces then CAT (0). Definition B.0.28. Let (X, d) be a CAT (0) metric space, and r ⊂ X be a ray. A Busemann function associated to the ray r is the function fr : X → R, fr (x) = lim ( d(x, r(t)) − t ). t→∞ If r1 and r2 are two asymptotic geodesic rays, then fr1 − fr2 is a constant function ([8]). Hence the family of level sets {x | fr (x) ≤ a} and the family of level hypersurfaces {x | fr (x) = a} of the Busemann function do not depend on ray r but only on its point at infinity α = r(∞). 104

A horoball centered in α is the level set of the Busemann function. One call a level hypersurface of the Busemann function horosphere centered in α. Finally, the set {x | fr (x) < a} is called open horoball centered in α. We denote Hb(r) the horoball {x | fr (x) ≤ 0} and we call it the horoball determined by the ray r. In the same way, H(r) = {x | fr (x) = 0}, is called the horosphere determined by r. Definition B.0.29. A lattice Γ is called arithmetic lattice if there exists a connected semi-simple, algebraic Q-group G0 and a R-epimorphism Φ : G0 → G such that Ker Φ is compact, Γ ⊂ Φ(G0 (Q)) and Γ is commensurable with Φ(G0 (Z)). Theorem B.0.30 (Margulis Arithmeticity Theorem). If the R-rank of G is greater then 1, then any irreducible lattice is G is arithmetic. Definition B.0.31. Let G be a semisimple group of R-rank at least 2 and Γ ⊂ G be an irreducible lattice. The rank over Q of the Q-group G0 , associated to Γ by the arithmeticity theorem, is called the Q-rank of the lattice Γ. Remark B.0.32. One can also define the Q-rank of Γ as the maximal dimension of subgroups of Γ0 diagonalizable over Q. Let now X = G/K be the symmetric space associated to a semisimple Lie group G, and let Γ be a non-uniform lattice. Denote Y = X/Γ. Up to finite index, one can suppose that Γ is torsion free, hence that Y is a locally symmetric space. The space Y has a finite number of ends, and each end is a cusp. Moreover, Y has a compact core, which is obtained as a quotient by Γ of a subspace X0 of X, constructed as a complementary of a uncountable family of open horospheres. The boundary of such a space X0 is the union of an uncountable family of horospheres. If Γ has Q-rank one, such a family of horospheres and horoballs can be done disjoint, in fact: Theorem B.0.33 (Raghunathan). Let Γ ⊂ G be a Q-rank 1, irreducible lattice of a semisimple Lie group. There exist r1 , r2 , . . . , rk rays in the symmetric space X associated to G, such that (1) Hb(γri ) ∩ Hb(rj ) = ∅, if i 6= j, γ ∈ Γ. (2) the space X0 = X −

k G G

Hbo(γri ) has a compact quotient X0 /Γ.

i=1 γ∈Γ

105

Definition B.0.34. The space X0 is called the neutered space associated to Γ. Now Γ acts properly discontinuously and co-compactly by isometries on X0 , giving a quasi-isometry of Γ with this neutered space X0 . Definition B.0.35. Let (X, d) be a metric space and A ⊂ X be a subset endowed with a metric dA . One says that (A, dA ) is undistorted in X if and only if there exists a constant L ≥ 0 such that 1 d(x, y) ≤ dA (x, y) ≤ Ld(x, y), ∀x, y ∈ A. L Theorem B.0.36 (Lubotzky–Mozes–Raghunathan). Let G be a semisimple Lie group of R-rank at least 2, and let Γ be an irreducible lattice in G. Then Γ with the word-metric is undistorted in G endowed with a left-invariant metric. For Q-rank 1 lattices, Γ with the word metric is quasi-isometric to the neutered space X0 with the length-metric. Hence, in order to show that (X0 , d` ) is undistorted in X, it suffices to show it for horospheres H(γri ), since horoballs Hb(γri ) are disjoint. Thus, for Q-rank 1 lattices, the Lubotzky–Mozes–Raghunathan theorem is equivalent to the non distortion of horospheres.

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REMINDER • Asymptotic property (or Geometric property): A property (∗) of finitely generated groups is said to be a geometric (or asymptotic) property if, whenever G1 and G2 are two quasi-isometric (finitely generated) groups, G1 has property (P ) if and only if G2 has property (∗). • CAT(0) space: Let X be a geodesic space. Given a triangle T in X, there is a triangle T ∗ in R2 with the same edge lengths. The triangle T is said to satisfy the CAT(0)-inequality, if, given any two points x, y ∈ T we have d(x, y) ≤ d(x∗ , y ∗ ). A space X is a CAT(0)-space if it is complete and if the CAT(0)-inequality holds for all triangles in X. • Cayley graph: The Cayley graph associated to a group G generated by a finite set S is the graph whose vertices are the elements of S, and for each s ∈ S and w ∈ G, there is an oriented edge labelled by s from w to ws. • Cayley 2-complex and standard complex associated to a presentation: Let P = (S, R) be a finite presentation of a group G. The standard 2-complex XP associated to P is constructed as follows. Starts with a bouquet, i.e. a vertex v and one edge loop at v for any s ∈ S oriented and labelled by s. Then for each relator r ∈ R attach an l(r)-sides 2-cell to the bouquet using r to describe the attaching map (here l(r) denotes the length of r). Then, π1 (XP ) = G, and its universal covering is the Cayley 2-complex of G. • Discrete groups: Topological groups with the discrete topology. Often in practise, discrete groups arise as discrete subgroups of continuous Lie groups acting on a geometric space, or as symmetries of discrete structures (e.g. graphs, lattices, polyhedra), or as fundamental groups. • Geometric simple connectivity (gsc): A (possibly non-compact) manifold is gsc if it admits a proper handlebody decomposition without handles of index 1 (handle decomposition are known to exist for all n -manifolds with n 6= 4). • Weak geometric simple connected (wgsc): A (non-compact) polyhedron P is wgsc if P = ∪i Ki where Ki are compact, simply connected polyhedra such that Ki ⊂ Ki+1 . Or, equivalently, any compact subset is contained in a simply connected compact subpolyhedron. 107

Similar definitions can be given in the case of topological (resp. smooth) manifolds where we require the exhaustion to be by topological (resp. smooth) submanifolds. • Quasi-isometry: A quasi isometry between two metric spaces (X, dX ) and (Y, dY ) is a map f : X → Y such that λ−1 dX (x1 , x2 )−C ≤ dY (f (x1 ), f (x2 )) 6 λdX (x1 , x2 )+C,

∀x1 , x2 ∈ X

for some fixed positive constants C and λ, and Y is at finite Hausdorff distance from the image of f (i.e. for any y ∈ Y there exists f (x) ∈ f (X) such that dY (y, f (x)) ≤ L). • Quasi simple filtration (qsf ): A polyhedron X is qsf if for any compact subset K ⊂ X there exists a compact, simply connected polyhedron YK and a map f : YK → X such that K ⊂ f (YK ) and the map f restricted to f −1 (K) is an omeomorphism between f −1 (K) and K. • Simple connectivity at infinity (sci): A space X is sci if for any compact subset k there exists another compact K such that any loop in X − K are null-homotopic outside k. • Tucker property: A compact complex X has the Tucker property (or e it is Tucker), if given any finite subcomplex C of its universal cover X, e − C) is finitely generated for any choice the fundamental group π1 (X e − C). of basepoint (i.e., for each component of X We end by giving some relations between some of the properties defined above: 1. Irreducible open 3-manifolds : sci ⇔ gsc ⇔ wgsc. 2. Open simply connected manifolds of dimension greater then 5: sci ⇒ gsc. While sci : gsc (or wgsc). 3. Dimension n 6= 4: a non-compact n-manifold W n is wgsc ⇔ W n is gsc.

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