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Boundaries of Groups the American Institute of Mathematics Boundaries of groups The American Institute of Mathematics The following compilation of participant contributions is only intended as a lead-in to the AIM workshop “Boundaries of groups.” This material is not for public distribution. Corrections and new material are welcomed and can be sent to [email protected] Version: Tue Oct 11 13:34:04 2016 1 2 Table of Contents A.ParticipantContributions . 3 1. Bell, Robert 2. Bourdon, Marc 3. Charney, Ruth 4. Cordes, Matt 5. Dani, Pallavi 6. Davis, Mike 7. Groves, Daniel 8. Guilbault, Craig 9. Hagen, Mark 10. Haissinsky, Peter 11. Manning, Jason 12. Melnick, Karin 13. Moran, Molly 14. Osajda, Damian 15. Paoluzzi, Luisa 16. Puls, Michael 17. Sathaye, Bakul 18. Sisto, Alessandro 19. Stark, Emily 20. Tshishiku, Bena 21. Xie, Xiangdong 3 Chapter A: Participant Contributions A.1 Bell, Robert I hope to learn more about the latest techniques in the study of boundaries of CAT(0) groups. My primary objective for this workshop is to identify specific projects to which I can contribute. A problem which I find particularly compelling is finding conditions which imply that certain topological properties hold for every CAT(0) boundary of a group. The well-know examples of Croke and Kleiner and a generalization due to Mooney show that once a group has many intersecting flats (e.g. the right-angled Artin group defined by the path of four vertices), then the group can act cocompactly, isometrically, and properly on two different CAT(0) spaces, whose boundaries are non-homeomorphic. On the other hand, as shown by Hruska and Kleiner, if a CAT(0) group has the isolated flats property then any two boundaries are homeomorphic. My background includes some expertise in the study of Artin groups, Coxeter groups, and mapping class groups. A.2 Bourdon, Marc I am interested in boundaries of Gromov hyperbolic groups. More precisely, I am interested in their quasi-conformal geometry, and to related quasi-isometric invariants like the conformal dimension and the Lp-cohomology. A.3 Charney, Ruth There have been numerous recent developments aimed at generalizing hyperbolic tech- niques to broader classes of groups, such as acylindrically hyperbolic groups. In my work with Harold Sultan, later generalized by Matt Cordes, we introduce a “Morse boundary” that identifies aspects of hyperbolic behavior in any geodesic metric space by restricting to rays satisfying a Morse property. I am interested in discussing properties and applications of the Morse boundaries for various classes of groups (see, for example, the work of Devin Mur- ray on CAT(0) groups) as well as its relation with other types of boundaries. For example, how are topological properties of the Morse boundary, such as local connectivity, reflected in the group? In addition, I would like to explore possible measures on these spaces and dynamical properties of the group action. A.4 Cordes, Matt I study the Morse boundary of groups, a quasi-isometry invariant boundary that is non- empty for spaces with some aspect of negative curvature, i.e., if there exists a Morse geodesic. Sisto showed that every acylindrically hyperbolic group has a Morse geodesic. (The class of acylindrically hyperbolic groups includes MCG(S), Out(Fn), relatively hyperbolic groups, non-directly decomposable right-angled Artin groups, and more.) In the case of a hyperbolic group, this boundary is the normal Gromov boundary. There are many open questions about this boundary. It would be interesting to understand the relationship between the Morse boundary and the many other boundaries for the groups listed above or related spaces. And if so, can we extract some interesting information from the understanding the relationship. During the workshop, I would be happy to learn about some of these other boundaries. 4 Another open area of study would be to understand more about the dynamical prop- erties of the Morse boundary. In the CAT(0) setting, Devin Murray has explored some of these results and it would be nice to see which generalize, which do not, and why. Furthermore, it would be interesting to investigate if there is a connection between the topology of the Morse boundary of a group and algebraic information about that group. (The topology can be pretty wild; Murray showed that the Morse boundary of the group Z ∗ Z2 does not have a locally countable basis!) A.5 Dani, Pallavi Recently I have been investigating the quasi-isometry classification of right-angled Cox- eter groups (RACGs)in collaboration with Anne Thomas. We used the (visual) boundaries of RACGs to give a description of the Bowditch JSJ tree in the class of 2-dimensional hyper- bolic RACGs for which this tree is defined, leading to a complete quasi-isometry classification of a large class of hyperbolic RACGs. I am interested in understanding Morse boundaries for RACGs, with a view to applying them to the q.i.-classification question. This should have broader appeal, as it will lead to large collection of tractable examples of Morse boundaries. I am also interested in comparing different boundaries for RACGs. A.6 Davis, Mike The action dimension of Artin groups The action dimension of a discrete group G is the smallest dimension of a contractible manifold on which it can act properly. Bestvina, Kapovich and Kleiner [bkk] have showed how to use van Kampen’s embedding obstruction to compute the action dimension of G. If G has a Z-set boundary, ∂G, and if K is a finite complex embedded in ∂G such that the van Kampen obstruction for embedding K in Sm is nonzero, then the action dimension of G is ≥ m +2. When G is not known to have a Z-set boundary, one replaces the condition that K ⊂ ∂G, by the condition that there is a uniform proper embedding of the 0-skeleton of Cone(K) into G. In [ados, dj] my collaborators and I used this idea to compute the action dimension in the case when G is an Artin group. Bibliography [ados] G. Avramidi, M.W. Davis, B. Okun and K. Schreve) The action dimension of right- angled Artin groups, Bull. London Math. Soc. 48 (1) (2016), 115-126. [bkk] M. Bestvina, M. Kapovich and B. Kleiner, Van Kampen’s embedding obstruction for discrete groups, Invent. Math. (2) 150 (2002), 219–235. [dj] M.W.Davis and J. Huang, Determining the action dimension of an Artin group by using its complex of abelian subgroups, arXiv:1608.03572. A.7 Groves, Daniel My recent interest in boundaries has arisen in joint work with Jason Manning and Manning and Sisto, where we begin to understand what happens to the (Bowditch) boundary of a relatively hyperbolic group under the operation of group-theoretic Dehn filling. This has lead to a study of the topological and analytic properties of these boundaries. I have a number of questions about boundaries that I think are interesting, and that I would like to know the answer to. Here are a few: 5 Question 1. Can the new theories of hyperbolicity, such as acylindrical hyperbolicity be characterized in terms of convergence group actions of some kind? Note that the Gromov boundary of many of the spaces arising in this theory are not compact. Question 2. Is there some way of using the boundary of a hyperbolic group to develop a group-theoretic notion of ‘drilling’ which complements the theory of relatively hyperbolic Dehn filling? Question 3. Suppose that G is a hyperbolic group such that ∂G has no local cut points. Is there some homeomorphism η[0, ∞) → [0, ∞) so that for any ǫ> 0 we have: A. there exist quasicircles in ∂G of quality η; and B. for some family F of such quasicircles (of quality η), each of which induce a quasi-isometry 2 H → G whose image comes uniformly close to 1 so that for any pair Q1,Q2 ∈ F there is a chain of quasicircles between Q1 and Q2 which are pairwise distance at most ǫ apart? Note: The distance between two quasicircles should come from integrating (around S1) 1 the distance between points Q1(t),Q2(t) for t ∈ S with respect to a visual metric on ∂G based at 0. I would be interested in this question in case ∂G is a 2–sphere. In this case, would such a condition already imply that ∂G has enough regularity so that the Cannon conjecture is true? [In which case we could take F to be the set of round circles whose convex hull includes a fixed basepoint of H2, interpreting the geometry of G as H3.] We remark that Bonk and Kleiner proved that all boundaries of one-ended hyperbolic groups admit some quasicircle and, under the assumption of no local cutpoints, Mackay proved that a quasicircle can be made to go through any finite collection of points. Neither of these seem like enough to answer the above question, which asks for many more quasicircles. A.8 Guilbault, Craig Group boundaries provide a bridge between the classical and ongoing study of ends of spaces and the subject of geometric group theory. From that perspective, I find Bestvina’s axiomatic approach to group boundaries particularly appealing. That approach is based on an established concept from topology—that of a Z-set. A closed subset A of a space B is a Z-set if U − A ֒→ U is a homotopy equivalence for every open U ⊆ B. For example, the b boundary of a manifold M n is a Z-set in that manifold. A compactification X = X ∪ Z of a b space X is a Z-compactification provided Z is a Z-set in X.
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