RESEARCH PROGRAM Daniele Ettore Otera

This research program is concerned with topological and geometric properties of groups whose de nitions are inspired by low-dimensional topology (at in nity). The central themes of the present program are the (simple) connectivity at in nity, the geometric simple connectivity and their invariance. Let me emphasize the dierence between the asymptotic invariants of groups of geometric nature and those of topological nature. Geometric invariants of groups are soft and very sensitive to cut and paste operations and thus nontrivial examples are easy to obtain by means of standard constructions in algebra. On the other side, topological properties are quite stable and thus, the quantitative measurements of these properties tend to be generically trivial. In particular, examples where the nontrivial topology is manifest quantitatively are much harder to obtain. This is precisely the case of the following properties. 0.1. The simple connectivity at in nity. One of the main invariant of noncompact spaces is the simple connectivity at in nity. A nitely presented group is said to be simply connected at in nity (and one writes sci ) if some (equivalently any) presentation complex X is simply connected at in nity, i.e. every compact K of X is contained in a compact L such that loops outside L can be lled outside K . The startpoint my PhD thesis was to show the quasi-isometry invariance of the simple connectivity at in nity for nitely presented groups; the results in [8] and [4] are actually much stronger and add a quantitative avour. In fact, we introduced a function V , called the sci growth, which is de ned for those metric spaces which are simply connected at in nity. Roughly speaking, one considers the smallest function V1 (r) such that loops outside the ball of radius V1 (r) are null-homotopic outside the ball of radius r. We proved then that the coarse equivalence class of the function V for the universal covering of a compact polyhedron with given fundamental group G is a quasi-isometry invariant of the group G. We obtained therefore a new quasi-isometry invariant for groups which are simply connected at in nity, which is interesting and will deserve further investigations. One should notice that this invariant is of topological nature and thus it is highly stable in comparison with those of geometric nature like the isodiametric or Dehn functions. Further, we computed this invariant and showed that the class of V1 is trivial (i.e. linear) for many cocompact lattices in connected Lie groups, in particular for semi-simple, nilpotent and (several classes of) solvable Lie groups. As a consequence, this holds true for all geometric 3-manifold groups and conjecturally for all connected Lie groups. Now, the next step is to consider non-uniform lattices in Lie groups. This is equivalent to the study of the asymptotic topology of a symmetric space from which an in nite collection of horoballs are deleted. Using a deep result of Lubotzky, Mozes and Ragunathan, I proved in the thesis that non-uniform lattices in Lie groups of R-rank at least 2 and Q-rank 1 have a linear sci growth. On the other hand one expects that there exist groups with nonlinear growth function V1 . The stability alluded above makes the construction of explicit examples to be highly nontrivial. 0.2. The end-depth. I approached the last problem from a dierent perspective. In fact, the same techniques can be used to prove that the higher - as well as the lower, namely the plain - connectivity at in nity, are also quasi-isometry invariants of groups and lead to quasi-isometry invariants Vk de ned for groups that are k-connected at in nity. The case when k is at least 2 might have less interest in the absence of new and interesting examples. But the case k = 0 (of the connectivity at in nity) leads to a genuinely new end invariant V0 that I baptised the end-depth. The main result in this direction is that amalgamated products of groups with nontrivial V0 should have nontrivial V1 (i.e. sci growth). This relationship between dierent (and independent) invariants seems to be most promising. I wish to emphasize the fact that the some specialists were interested in the last few years in nding groups with in nite end-depth functions (namely the work of Cannon, Guba, Cleary and Riley about the dead-end elements and their depth). It appears that this property (e.g. have an unbounded dead-end depth) depends on the particular group presentation and, if we wish something to be well-de ned, then we are forced to consider the coarse equivalence class of the end-depth, namely the class V0 . Thus in some sense the end-depth provides the right framework to study the connectivity at in nity. The previous known examples still furnish groups with linear V0 . November 2006. 1

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The results above are part of a more general philosophy based on the following conjectural statement: if a group is word hyperbolic (or quasi-isometric to a CAT(0) space) and simply connected at in nity (notice that there are hyperbolic groups which are not simple connected at in nity) then it has linear sci growth. The analog result for V0 holds true (see [8], where one can also nd a proof of the previous statement for Coxeter, Artin groups and euclidean buildings). The construction of groups with nontrivial end-depth is our main goal, even if there exist some doubts about the linearity of V0 for any nitely generated group. 0.3. The weak geometric simple connectivity. The second part of this program looks to another topological notion relevant for noncompact spaces, that of the geometric simple connectivity, which was developed in the papers of Poenaru (mostly in dimensions 3 and 4, in his work concerning the Poincare Conjecture [9]) and then by Gadgil and Funar ((in higher dimensions as well [3]). Roughly speaking, geometric simple connectivity means that there are no 1-handles in a proper handelobody decomposition and this makes sense only for simply connected spaces. There is a natural extension of the geometric simple connectivity to the realm of polyhedra: a noncompact polyehdron is weak geometrically simply connected if it can be written as an ascending union of compact simply connected subpolyhedra. The second part of my thesis (as well as [5]) claried several aspects of the relationship between the geometric simple connectivity and some other topological tameness conditions of non-compact spaces, namely the quasi simple ltration property studied by Brick, Mihalik and Stallings ([10] and the Dehn-exhaustibility introduced by Poenaru [9]. First, we proved that these notions are equivalent for groups and for spaces which are manifolds of dimension at least 5. Moreover, we obtained a number of examples (one-relator groups, sci groups, almost-convex groups, etc), and of constructions leading to wgsc groups: using extensions, amalgamated free products along nitely generated subgroups, etc. In a few words, all known examples of groups are weakly geometrically simply connected (it is likely that all diagram groups considered by Guba and Sapir are wgsc; and one might wonder whether nitely presented groups that have solvable word problem are actually wgsc). The examples from [3] might suggest that non wgsc Cayley complexes (i.e. nitely presented groups) should also exist. Discrete groups which are not wgsc (if they exist) would lay at the opposite extreme to hyperbolic (or non-positively curved, i.e. acting cocompactly on CAT(0) spaces) groups and thus they should be non generic, in the probabilistic sense. It is hard to believe that all nitely presented groups could be wgsc, but there are still not known explicit examples. A more dicult problem is whether one could nd aspherical non wgsc manifolds which are universal coverings of compact polyhedra. We have re ned the wgsc property by introducing the wgsc growth function fG , which is the rough equivalence class of the function fG (r) = inf R such that the ball of radius R contains a compact, simply-connected subspace containing B (r). However, it is still unknown whether the rough equivalence class of fG is a quasi-isometry invariant. Furthermore, we showed that, for a wgsc group, requiring that the exhaustion be that by metric balls in a Cayley complex puts already strong constraints on the group and thus it does not hold for general wgsc groups. In fact, if some Cayley complex of a nitely presented group has balls (or spheres) whose 1 's are generated by uniformly small loops, then the group is wgsc with linear wgsc growth. The work done in [5] shows up other classes of groups which are wgsc, namely the Thompson groups and the Grigorchuk HNN extension (which is nitely presented) of his well-known torsion intermediate growth automaton group. Such results confort the idea that, searching for non wgsc groups, one should consider much larger groups (like Burnside ones) in the kernel: our study suggests that similar but more involved construction might lead to non wgsc groups, which will answer to a question of Stallings. 0.4. Further perspectives. A nal interesting aspect to develop is the extension of some results by Funar ([2]) about the simple connectivity at in nity and the geometric simple connectivity in dimension 3, for 3-manifolds with boundary. In such a case, the gsc condition does not imply the simple connectivity at in nity, but one could (should) prove that the gsc implies that the fundamental group at in nity is free abelian (the easiest case of non-trivial fundamental group at in nity). First of all, one should extend the wgsc notion to the case with boudary. An interesting condition to study and exploit is that of properly 3-realizable (P3R) groups ([1]), which means that the Cayley complex of the group is properly homotopy equivalent to a 3-manifold (possibly non-open). An interesting subject to explore is the relations between the wgsc, the P3R and the semistability at in nity (see [7]) in the realm of groups.

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References [1] M. Cardenas and F.Lasheras, Properly 3-realizable groups: a survey, Geometric methods in group theory, 1-9, Cont.Math. 372, Amer. Math. Soc., Providence, RI, 2005. [2] L.Funar, On proper homotopy type and the simple connectivity at in nity of open 3-manifolds, Atti Sem. Mat. Fis. Univ. Modena, 49 (2001), 15{29. [3] L.Funar and S.Gadgil, On the geometric simple connectivity of open manifolds, I.M.R.N. 2004, n. 24, 1193-1248. [4] L.Funar and D.E.Otera, A re nement of the simple connectivity at in nity of groups, Archiv Math. (Basel) 81(2003), 360-368. [5] L.Funar and D.E.Otera, Remarks on the wgsc and qsf tameness conditions for groups, math.GT/0610936, October 2006. [6] M.Gromov, Asymptotic invariants of in nite groups, Geometric group theory, Vol. 2 (Sussex, 1991), 1{295, London Math. Soc. Lecture Note Ser., 182, Cambridge Univ. Press, Cambridge, 1993. [7] M.Mihalik, Semistability at , -ended groups and group cohomology, Trans. Amer. Math. Soc. 303 (1987), no. 2, 479{485. [8] D.E.Otera, Asymptotic Topology of Groups. Connectivity at in nity and geometric simple connectivity, PhD Thesis, Universita di Palermo and Universite de Paris-Sud, 02-2006. [9] V.Poenaru, Killing handles of index one stably and 1 , Duke Math. J. 63(1991), 431{447. [10] J.R.Stallings, Brick's quasi-simple ltrations for groups and 3-manifolds, Geometric group theory, Vol. 1 (Sussex, 1991), 188{203, London Math. Soc. Lecture Note Ser., 181, Cambridge Univ. Press, Cambridge, 1993.

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