Actions of certain arithmetic groups on Gromov hyperbolic spaces

Manning, Jason Fox

doi:10.2140/agt.2008.8.1371

Cited by 9 publications

(14 citation statements)

References 26 publications

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“…Remark 4.19. By [48,Lemma 4.5], |q x (g) − (d S (gs, x n ) − d S (s, x n ))| is uniformly bounded when g ranges in G and hence so is |p(g) − (d S (gs, x n ) − d S (s, x n ))| (see (15)). Proof.…”

Section: Lineal Hyperbolic Structures and Pseudocharactersmentioning

confidence: 99%

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Hyperbolic structures on groups

Abbott

Balasubramanya

Osin

2019

Algebr. Geom. Topol.

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: Lineal Hyperbolic Structures and Pseudocharactersmentioning

confidence: 99%

“…Assume first that [Y ] is lineal, i.e., T is quasi-isometric to a line. Then (48) implies that (a i ) → ξ ∈ ∂T (possibly after passing to a subsequence). Let x = (a i ), s = 1, and let q x and p be the corresponding Busemann quasi-character and pseudocharacter, see Definition 4.18.…”

Section: Coarsely Isospectral Actionsmentioning

confidence: 99%

Hyperbolic structures on groups

Abbott

Balasubramanya

Osin

2019

Algebr. Geom. Topol.

110

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“…Its homogenization p x is called the Busemann pseudocharacter. It is known that this definition is independent of the choice of x (see [5,Lemma 4.6]) and thus we can drop the subscript in p x . It is straightforward to verify that g ∈ G acts loxodromically on S if and only if p(g) = 0; in particular, p is non-zero whenever G S is quasi-parabolic.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic structures on wreath products

Balasubramanya

2019

Journal of Group Theory

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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?

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“…is finite; it is called homogeneous if moreover f (g n ) = n f (g) for all g ∈ G and n ∈ Z; in that case, f is constant on conjugacy classes. Given an isometric group action on X fixing ξ, there is a canonical homogeneous quasicharacter associated to the action, which was constructed by J. Manning [Man08,Sec. 4].…”

Section: Proposition 32 If the Action Of γ Is Bounded Lineal Or Fomentioning

confidence: 99%

Amenable hyperbolic groups

Caprace¹,

Cornulier²,

Monod³

et al. 2015

J. Eur. Math. Soc.

194

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Abstract. We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic locally compact groups with a cusp-uniform non-uniform lattice, is very restricted.

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Actions of certain arithmetic groups on Gromov hyperbolic spaces

Abstract: We study the variety of actions of a fixed (Chevalley) group on arbitrary geodesic, Gromov hyperbolic spaces. In high rank we obtain a complete classification. In rank one, we obtain some partial results and give a conjectural picture.

Cited by 9 publications

References 26 publications

Hyperbolic structures on groups

Hyperbolic structures on groups

Hyperbolic structures on wreath products

Amenable hyperbolic groups

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