Extending group actions on metric spaces

Abbott, Carolyn R.; Hume, David; Osin, Denis

doi:10.1142/s1793525319500584

Cited by 7 publications

(13 citation statements)

References 18 publications

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“…Let H be a subgroup of a group G and let X be a relative generating set of G with respect to H. That is, G = X ∪ H . It is easy to see that the map sending a generating set Y of H to X ∪ Y gives rise to a map ι X : G(H) → G(G), which can be thought of as a particular case of the induced action map studied in [3]. In general, very little can be said about ι X .…”

Section: 2mentioning

confidence: 99%

“…Finally we recall the definition of equivalent group actions introduced in [3]. Two actions G R and G S are equivalent, denoted G R ∼ G S, if there exists a coarsely G-equivariant quasi-isometry R → S. It is easy to prove (see [3]) that ∼ is indeed an equivalence relation.…”

Section: Comparing Group Actionsmentioning

confidence: 99%

“…We begin by recalling some useful results from [3], which play the central role in the proof of Theorems 2.6, 2.9, and 2.11. Let H 1 , .…”

Section: Acylindricity Of Induced Structuresmentioning

confidence: 99%

“…This map can be thought of as the analogue of the induced action map defined in [3] for equivalence classes of group actions on geodesic metric spaces. In the theorem below, we restate some of the results of [3] using terminology of this paper.…”

Section: Acylindricity Of Induced Structuresmentioning

confidence: 99%

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Hyperbolic structures on groups

Abbott

Balasubramanya

Osin

2019

Algebr. Geom. Topol.

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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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Section: 2mentioning

confidence: 99%

Section: Comparing Group Actionsmentioning

confidence: 99%

“…We begin by recalling some useful results from [3], which play the central role in the proof of Theorems 2.6, 2.9, and 2.11. Let H 1 , .…”

Section: Acylindricity Of Induced Structuresmentioning

confidence: 99%

Section: Acylindricity Of Induced Structuresmentioning

confidence: 99%

See 2 more Smart Citations

Hyperbolic structures on groups

Abbott

Balasubramanya

Osin

2019

Algebr. Geom. Topol.

Self Cite

110

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“…The latter theorem is especially useful in conjunction with various "extension" results proved in [1,29,40]. Roughly speaking, these results claim that various things (e.g., group actions on metric spaces or quasi-cocycles) can be "extended" from a hyperbolically embedded subgroup to the whole group.…”

Section: Hyperbolically Embedded Subgroups In Acylindrically Hyperbolmentioning

confidence: 90%

Groups Acting Acylindrically on Hyperbolic Spaces

Osin

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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The goal of this article is to survey some recent developments in the study of groups acting on hyperbolic spaces. We focus on the class of acylindrically hyperbolic groups and their hyperbolically embedded subgroups. This class is broad enough to include many examples of interest, yet a significant part of the theory of hyperbolic and relatively hyperbolic groups can be generalized in this context.2 Acylindrically hyperbolic groups 2.1. Hyperbolic spaces and group actions. We begin by recalling basic definitions and general results about groups acting on hyperbolic spaces. Our main reference is the Gromov's paper [33]; additional details can be found in [12] and [31]. All group actions on metric spaces discussed in this paper are assumed to be isometric by default.Definition 2.1. A metric space S is hyperbolic if it is geodesic and there exists δ ≥ 0 such that for any geodesic triangle ∆ in S, every side of ∆ is contained in the union of the δ-neighborhoods of the other two sides.Example 2.2. Every bounded space S is hyperbolic with δ = diam(S). Every tree is hyperbolic with δ = 0. H n is hyperbolic for every n ∈ N. On the other hand, R n is not hyperbolic for n ≥ 2.Given a hyperbolic space S, we denote by ∂S its Gromov boundary. We do not assume that the space is proper and therefore the boundary is defined as the set of equivalence

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Actions of small cancellation groups on hyperbolic spaces

Abbott

Hume

2020

Geom Dedicata

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Extending group actions on metric spaces

Cited by 7 publications

References 18 publications

Hyperbolic structures on groups

Hyperbolic structures on groups

Groups Acting Acylindrically on Hyperbolic Spaces

Actions of small cancellation groups on hyperbolic spaces

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