On acylindrical hyperbolicity of groups with positive first ℓ<sup>2</sup>-Betti number

Osin, Denis

doi:10.1112/blms/bdv047

Cited by 20 publications

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“…A group is called acylindrically hyperbolic if it is not virtually cyclic and admits an acylindrical action on a hyperbolic metric space with unbounded orbits [49]. The class of acylindrically hyperbolic groups includes many examples of interest; for an extensive list we refer to [15,41,49,48].…”

Section: Introductionmentioning

confidence: 99%

Transitivity degrees of countable groups and acylindrical hyperbolicity

Hull¹,

Osin²

2016

Isr. J. Math.

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We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive faithful actions of mapping class groups. It also implies that in various geometric and algebraic settings, the transitivity degree of an infinite group can only take two values, namely 1 and ∞. Here by transitivity degree of a group we mean the supremum of transitivity degrees of its faithful permutation representations. Further, for any countable group G admitting a highly transitive faithful action, we prove the following dichotomy: Either G contains a normal subgroup isomorphic to the infinite alternating group or G resembles a free product from the model theoretic point of view. We apply this theorem to obtain new results about universal theory and mixed identities of acylindrically hyperbolic groups. Finally, we discuss some open problems.

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Section: Introductionmentioning

confidence: 99%

Transitivity degrees of countable groups and acylindrical hyperbolicity

Hull¹,

Osin²

2016

Isr. J. Math.

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“…An application to the Baum-Connes conjecture is studied by W. Lück in [77]. Relation between the first L 2 -Betti numbers and acylindrical hyperbolicity of groups is investigated by D. Osin [93]. Also D. Osin [92] constructed the first examples of finitely generated, nonunitarizable groups without nonabelian free subgroups using L 2 -Betti numbers.…”

Section: Applications and Motivationsmentioning

confidence: 99%

L2-Betti Numbers and their Analogues in Positive Characteristic

Jaikin–Zapirain¹

2019

Groups St Andrews 2017 in Birmingham

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Section: Introductionmentioning

confidence: 99%

“…In particular, acylindrically hyperbolic groups share many interesting properties with non-elementary hyperbolic and relatively hyperbolic groups. For details we refer to [5,10,11,12] and references therein.The main goal of this paper is to answer the following.Question 1.1. Which groups admit non-elementary cobounded acylindrical actions on quasi-trees?In this paper, by a quasi-tree we mean a connected graph which is quasi-isometric to a tree.…”

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confidence: 99%

Acylindrical group actions on quasi-trees

Balasubramanya

2017

Algebr. Geom. Topol.

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A group G is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that every acylindrically hyperbolic group G has a generating set X such that the corresponding Cayley graph Γ is a (non-elementary) quasi-tree and the action of G on Γ is acylindrical. Our proof utilizes the notions of hyperbolically embedded subgroups and projection complexes. As an application, we obtain some new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups. IntroductionRecall that an isometric action of a group G on a metric space (S, d) is acylindrical if for every ε > 0 there exist R, N > 0 such that for every two points x, y with d(x, y) ≥ R, there are at most N elements g ∈ G satisfying d(x, gx) ≤ ε and d(y, gy) ≤ ε.Obvious examples are provided by geometric (i.e., proper and cobounded) actions; note, however, that acylindricity is a much weaker condition.A group G is called acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. Over the last few years, the class of acylindrically hyperbolic groups has received considerable attention. It is broad enough to include many examples of interest, e.g., non-elementary hyperbolic and relatively hyperbolic groups, all but finitely many mapping class groups of punctured closed surfaces, Out(F n ) for n ≥ 2, most 3-manifold groups, and finitely presented groups of deficiency at least 2. On the other hand, the existence of a non-elementary acylindrical action on a hyperbolic space is a rather strong assumption, which allows one to prove non-trivial results. In particular, acylindrically hyperbolic groups share many interesting properties with non-elementary hyperbolic and relatively hyperbolic groups. For details we refer to [5,10,11,12] and references therein.The main goal of this paper is to answer the following.Question 1.1. Which groups admit non-elementary cobounded acylindrical actions on quasi-trees?In this paper, by a quasi-tree we mean a connected graph which is quasi-isometric to a tree. Quasi-trees form a very particular subclass of the class of all hyperbolic spaces. From the asymptotic point of view, quasi-trees are exactly "1-dimensional hyperbolic spaces".

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On acylindrical hyperbolicity of groups with positive first ℓ²-Betti number

Cited by 20 publications

References 21 publications

Transitivity degrees of countable groups and acylindrical hyperbolicity

Transitivity degrees of countable groups and acylindrical hyperbolicity

L2-Betti Numbers and their Analogues in Positive Characteristic

Acylindrical group actions on quasi-trees

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On acylindrical hyperbolicity of groups with positive first ℓ2-Betti number

Cited by 20 publications

References 21 publications

Transitivity degrees of countable groups and acylindrical hyperbolicity

Transitivity degrees of countable groups and acylindrical hyperbolicity

L2-Betti Numbers and their Analogues in Positive Characteristic

Acylindrical group actions on quasi-trees

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On acylindrical hyperbolicity of groups with positive first ℓ²-Betti number