Hyperbolic structures on groups

Abstract

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.

Extending group actions on metric spaces

“…Thus one can get interesting actions of G on hyperbolic spaces starting from actions of F 2 and applying Theorem 1.12 (and Example 1.4 (b)). This idea is used in [1] to obtain several results about hyperbolic structures on groups.…”

Correction to: Acylindrical hyperbolicity of groups acting on trees

“…In fact, it is also possible to prove the above mentioned results of [2] for fundamental groups of closed (but possibly non-orientable) manifolds using tools from [1]. Let us briefly recall the necessary notation and terminology introduced in [1].…”

“…It is easy to show (see Lemma 5.23 in [1]) that the formula α([X ]) = [α(X )] for all α ∈ Aut (H ) and [X ] ∈ AH(H ) gives a well-defined order preserving action of Aut (H ) on AH(H ). From now on, let [X ] ∈ AH(H ) denote the largest element.…”

Acylindrical hyperbolicity and Artin-Tits groups of spherical type

“…Given a group G and a generating set X of G, the word metric d X turns G into a metric space; this space is (1, 1)-quasi-isometric to Γ(G, X), the Cayley graph of G with respect to X , endowed with the usual graph metric where each edge is identified to an interval of length 1. Hyperbolic structures on groups were recently introduced and studied in [1]. A hyperbolic structure on a group G is a generating set X of G such that (G, d X ) is Gromov-hyperbolic; note that X must be infinite whenever G is not itself hyperbolic.…”

“…We then study the relationships between X A P , X A NP and X A abs . Following [1], given two generating sets X, Y of a group G, we write X Y if the identity map from (G, d Y ) to (G, d X ) is Lipschitz (or equivalently, if sup y∈Y d X (1 G , y) < ∞). The sets X and Y are equivalent if both X Y and Y X hold (or equivalently, if the identity map is a bilipschitz equivalence between (G, d X ) and (G, d Y )).…”