Groups Acting Acylindrically on Hyperbolic Spaces

Osin, Denis

doi:10.1142/9789813272880_0082

Cited by 22 publications

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References 83 publications

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“…Recall that an isometric action of a group G on a metric space (S, d) is acylindrical [15] if for every constant ε there exist constants R = R(ε) and N = N (ε) such that for every x, y ∈ S satisfying d(x, y) ≥ R,Groups acting acylindrically on hyperbolic spaces have received a lot of attention in the recent years. For a brief survey we refer to [64].The goal of our paper is to initiate the study of the posets H(G) and AH(G) for various groups G and suggest directions for the future research. Our main results are discussed in the next section.…”

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confidence: 99%

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Hyperbolic structures on groups

Abbott

Balasubramanya

Osin

2019

Algebr. Geom. Topol.

110

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For every group G, we define the set of hyperbolic structures on G, denoted H(G), which consists of equivalence classes of (possibly infinite) generating sets of G such that the corresponding Cayley graph is hyperbolic; two generating sets of G are equivalent if the corresponding word metrics on G are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded G-actions on hyperbolic spaces. We are especially interested in the subset AH(G) ⊆ H(G) of acylindrically hyperbolic structures on G, i.e., hyperbolic structures corresponding to acylindrical actions. Elements of H(G) can be ordered in a natural way according to the amount of information they provide about the group G. The main goal of this paper is to initiate the study of the posets H(G) and AH(G) for various groups G. We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several results about hyperbolic structures induced from hyperbolically embedded subgroups of G, and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.We denote the set of hyperbolic structures by H(G) and endow it with the order induced from G(G).Since hyperbolicity of a space is a quasi-isometry invariant, the definition above is independent of the choice of a particular representative in the equivalence class [X]. Using the standard argument from the proof of the Svarc-Milnor Lemma, it is easy to show that elements of H(G) are in one-to-one correspondence with equivalence classes of cobounded actions of G on hyperbolic spaces considered up to a natural equivalence: two actions G S and G T are equivalent if there is a coarsely G-equivariant quasi-isometry S → T .We are especially interested in the subset of acylindrically hyperbolic structures on G, denoted AH(G), which consists of hyperbolic structures [X] ∈ H(G) such that the action of G on the corresponding Cayley graph Γ(G, X) is acylindrical. Recall that an isometric action of a group G on a metric space (S, d) is acylindrical [15] if for every constant ε there exist constants R = R(ε) and N = N (ε) such that for every x, y ∈ S satisfying d(x, y) ≥ R,Groups acting acylindrically on hyperbolic spaces have received a lot of attention in the recent years. For a brief survey we refer to [64].The goal of our paper is to initiate the study of the posets H(G) and AH(G) for various groups G and suggest directions for the future research. Our main results are discussed in the next section. Some open problems are collected in 8.Acknowledgements. The authors would like to thank Spencer Dowdall for useful conversations about mapping class groups, and Henry Wilton, Dani Wise and Hadi Bigdely for helpful discussions of 3-manifold groups. The authors also thank the anonymous referee for useful comments.

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confidence: 99%

“…Groups acting acylindrically on hyperbolic spaces have received a lot of attention in the recent years. For a brief survey we refer to [64].…”

mentioning

confidence: 99%

Hyperbolic structures on groups

Abbott

Balasubramanya

Osin

2019

Algebr. Geom. Topol.

110

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“…The class of such groups is large enough to admit a great number of groups of classical interest. Examples include non-(virtually cyclic) groups that are hyperbolic relative to proper subgroups, many 3-manifold groups, many groups acting on trees, non-(virtually cyclic) groups acting properly on proper CAT(0)-spaces and containing rank-one elements, non-cyclic directly indecomposable right-angled Artin groups, all but finitely many mapping class groups, groups of deficiency at least 2, Out(F n ) for n ≥ 2 (see [46,48] for references and historical remarks).…”

Section: Introductionmentioning

confidence: 99%

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Bogopolski

Corson

2021

Math. Ann.

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We describe homomorphisms ϕ : H → G for which G is acylindrically hyperbolic and H is a topological group which is either completely metrizable or locally countably compact Hausdorff. It is shown that, in a certain sense, either the image of ϕ is small or ϕ is almost continuous. We also describe homomorphisms from the Hawaiian earring group to G as above. We prove a more precise result for homomorphisms ϕ : H → Mod( ), where H is as above and Mod( ) is the mapping class group of a connected compact surface . In this case there exists an open normal subgroup V ≤ H such that ϕ(V ) is finite. We also prove the analogous statement for homomorphisms ϕ : H → Out(G), where G is a one-ended hyperbolic group. Some automatic continuity results for relatively hyperbolic groups and fundamental groups of graphs of groups are also deduced. As a by-product, we prove that the Hawaiian earring group is acylindrically hyperbolic, but does not admit any universal acylindrical action on a hyperbolic space.

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“…In this paper, we are interested in Gromov's hyperbolic groups as well as (strongly) relatively hyperbolic groups and the recent acylindrically hyperbolic groups. Proving that a group satisfies some hyperbolicity is very convenient as it provides interesting information of the group; see [GdlH90,Osi06,Osi17] and references therein for more information. However, it may be a difficult task to show that a given group actually has a negatively-curved behavior, motivating the need of general criteria.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolicities in CAT(0) cube complexes

Genevois

2020

Enseign. Math.

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This paper is a survey dedicated to the following question: given a group acting on a CAT(0) cube complex, how to exploit this action to determine whether or not the group is Gromov / relatively / acylindrically hyperbolic? As much as possible, the different criteria we mention are illustrated by applications. We also propose a model for universal acylindrical actions of cubulable groups, and give a few applications to Morse, stable and hyperbolically embedded subgroups. arXiv:1709.08843v2 [math.GR] 26 Apr 2019 PreliminariesA cube complex is a CW complex constructed by gluing together cubes of arbitrary (finite) dimension by isometries along their faces. It is nonpositively curved if the link of any of its vertices is a simplicial flag complex (ie., n + 1 vertices span a n-simplex if and only if they are pairwise adjacent), and CAT(0) if it is nonpositively curved and simply-connected. See [BH99, page 111] for more information. Proposition 2.2. [Gen16b, Proposition 2.9] Let X be a CAT(0) cube complex and A, B ⊂ X two convex subcomplexes. The projection proj B (A) is a geodesic subcomplex of B. Moreover, the hyperplanes intersecting proj B (A) are precisely those which intersect both A and B. Lemma 2.3. [HW08, Lemma 13.8] Let X be a CAT(0) cube complex, Y ⊂ X a convex subcomplex and x ∈ X a vertex. Any hyperplane separating x from its projection onto Y must separate x from Y . Lemma 2.4. [Gen16c, Proposition 2.6] Let X be a CAT (0) cube complex, C ⊂ X a convex subcomplex and x, y ∈ X two vertices. The hyperplanes separating the projections of x and y onto C are precisely the hyperplanes separating x and y which intersect C.Lemma 2.5. [HW08, Corollary 13.10] Let X be a CAT(0) cube complex and Y 1 , Y 2 ⊂ X two convex subcomplexes. If x ∈ Y 1 and y ∈ Y 2 are two vertices minimising the distance between Y 1 and Y 2 , then the hyperplanes separating x and y are precisely the hyperplanes separating Y 1 and Y 2 .Cycles of subcomplexes. Given a CAT(0) cube complex, a cycle of subcomplexes is a sequence of subcomplexes (C 1 , . . . , C r ) such that, for every i ∈ Z/rZ, the subcomplexes C i and C i+1 intersects.Proposition 2.7. Let (A, B, C, D) be a cycle of four convex subcomplexes. There exists a combinatorial isometric embedding [0, p] disjoint from B and D (resp. A and C).Proof. First of all, let us record a statement which is contained into the proof of [Gen17b, Proposition 2.111] (in the context of quasi-median graphs, a class of graphs including median graphs, i.e., one-skeleta of CAT(0) cube complexes).Fact 2.8. If a is a vertex of A ∩ D minimising the distance to B ∩ C and if b (resp. c, d) denotes the projection of a onto B (resp. B ∩ C, C), then there exists a combinatorial isometric embedding [0, p] × [0, q] → X such that (0, 0) = a, (p, 0) = b, (p, q) = c and (0, q) = d.By convexity of A, B, C and D, this implies that [0, p] Let J be a hyperplane intersecting [0, p] × {0}. We know from Lemma 2.3 that J must be disjoint from B. Moreover, if J intersects D, then it follows from Helly's property, satisfied by...

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Groups Acting Acylindrically on Hyperbolic Spaces

Cited by 22 publications

References 83 publications

Hyperbolic structures on groups

Hyperbolic structures on groups

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

Hyperbolicities in CAT(0) cube complexes

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