Hyperbolic structures on wreath products

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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“…The inequality (7) also implies that G Γ ∼ w G S. Taking X ∈ σ([G Γ]) and applying Proposition 3.12 completes the proof.…”

Section: Cobounded Actions and The Svarc-milnor Mapmentioning

confidence: 72%

“…Define a map φ : Γ → S as follows: for each vertex u of Γ, let φ(u) = u, and for each edge e of Γ, let φ(e) be a geodesic in S from φ(e − ) to φ(e + ). Then (7) implies that φ is a quasi-isometric embedding, and so by [19, Theorem III.H.1.9], Γ is hyperbolic.…”

Section: Cobounded Actions and The Svarc-milnor Mapmentioning

confidence: 99%

Hyperbolic structures on groups

Abbott

Balasubramanya

Osin

2019

Algebr. Geom. Topol.

110

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“…Nonetheless, significant progress has recently been made in describing the cobounded hyperbolic actions of families of classically studied solvable groups. The second author initiated this study by giving a complete description of the cobounded hyperbolic actions of the lamplighter groups (Z/nZ) Z for n ≥ 2 in [5]. The first and third authors then completely described the hyperbolic actions of Anosov mapping torus groups in [4] and the hyperbolic actions of solvable Baumslag-Solitar groups in [3].…”

Section: Introductionmentioning

confidence: 99%

“…The first two authors and Osin showed in [1] that the set H(G) of equivalence classes of cobounded hyperbolic actions of a group G admits a partial order, which roughly corresponds to collapsing equivariant families of subspaces to obtain one hyperbolic action from another (see Section 2 for the precise definition). Each of the papers [5], [3], and [4] gives a complete description of the poset H(G) for the group G in question.…”

Section: Introductionmentioning

confidence: 99%

“…The starting point for the techniques developed in [5,3,4] is a theory developed by Caprace, Cornulier, Monod, and Tessera in [7] that describes the cobounded hyperbolic actions of certain groups with rank one abelianizations, that is, groups whose abelianizations are virtually infinite cyclic. This includes groups G = H Z, with Z corresponding to a finite index subgroup of the abelianization.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Higher rank confining subsets and hyperbolic actions of solvable groups

Abbott¹,

Balasubramanya²,

Rasmussen³

2021

Preprint

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Recent papers of the authors have completely described the hyperbolic actions of several families of classically studied solvable groups. A key tool for these investigations is the machinery of confining subsets of Caprace, Cornulier, Monod, and Tessera, which applies, in particular, to solvable groups with virtually cyclic abelianizations. In this paper, we extend this machinery and give a correspondence between the hyperbolic actions of certain solvable groups with higher rank abelianizations and confining subsets of these more general groups. We then apply this extension to give a complete description of the hyperbolic actions of a family of groups related to Baumslag-Solitar groups.

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