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.

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".

The study of the poset of hyperbolic structures on a group G was initiated in [1]. However, this poset is still very far from being understood and several questions remain unanswered. In this paper, we give a complete description of the poset of hyperbolic structures on the lamplighter groups Z n wr Z and obtain some partial results about more general wreath products. As a consequence of this result, we answer two open questions regarding quasi-parabolic structures from [1]: we give an example of a group G with an uncountable chain of quasi-parabolic structures and prove that the Lamplighter groups Z n wr Z all have finitely many quasi-parabolic structures.Out of these, lineal and general-type actions were well-examined in [1], and several interesting examples and results were obtained. Of special interest were the following results: given any n ∈ N, there exist (distinct) finitely generated groups G n and H n such that |H (G n )| = n and |H gt (H n )| = n.However, the understanding of quasi-parabolic structures is far from being complete. It was shown in [1, Proposition 4.27] that Z wr Z has an uncountable antichain of quasi-parabolic structures, but little else is known. The authors of [1] consequently posed the following two open questions. Problem 1.1. Does there exist a group G such that H qp (G) is non-empty and finite? Problem 1.2. Does there exist a group G such that H qp (G) contains an uncountable chain?

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.

We prove that the group properties of being H−accessible and AH-accessible are preserved under finite extensions. We thus answer an open question from [1]. IntroductionAn important open question related to the class of acylindrically hyperbolic groups is the following (See [9, Question 2.20]).Question 1.1. Is the property of being acylindrically hyperbolic preserved under quasiisometries for finitely generated groups? In other words, if G is a finitely generated acylindrically hyperbolic group, and H is a finitely generated group that is quasi-isometric to G, then is H also an acylindrically hyperbolic group ?A group G is called acylindrically hyperbolic if it admits a non-elementary, acylindrical action on a hyperbolic space. The motivation behind the following question comes from the observation that the class of acylindrically hyperbolic groups serves as a generalization to the classes of non-elementary hyperbolic and relatively hyperbolic groups. The latter two classes are preserved under quasi-isometries (see [6]and [5, Theorem 5.12]), making Question 1.1 a natural question to consider.An answer to this question seems currently out of reach given the lack of techniques to build an action of H on a hyperbolic space simply starting from an action of G on a (possibly different) hyperbolic space and a quasi-isometry between the two groups. Indeed, this is hard to do even in the case of actions on Cayley graphs that arise from finite generating sets. By [10, Theorem 1.2], if G is acylindrically hyperbolic, then there exists a hyperbolic Cayley graph, say Γ(G, A), such that G Γ(G, A) is acylindrical and non-elementary. If G is acylindrically hyperbolic, but not a hyperbolic group, then A is necessarily infinite. Suppose that the groups G and H are quasi-isometric with respect to some finite generating sets X and Y respectively. Then the lack of a relationship between G Γ(G, X) and G Γ(G, A) makes it unclear as to how we may transfer the desired properties to the group H.Since the answer to this general question is out of reach, we may consider some special cases of Question 1.1 instead. For instance, we may ask the following.