A model for the universal space for proper actions of a hyperbolic group (original) (raw)
On the hyperbolic limit points of groups acting on hyperbolic spaces
Rendiconti del Circolo Matematico di Palermo, 1998
We study the hyperbolic limit points of a group G acting on a hyperbolic metric space, and consider the question of whether any attractive limit point corresponds to a unique repulsive limit point. In the special case where G is a (non-elementary) finitely generated hyperbolic group acting on its Cayley graph, the answer is affirmative, and the resulting map g+ ~---> g-is discontinuous everywhere on the hyperbolic boundary. We also provide a direct, combinatorial proof in the special case where G is a (non-abelian) free group of finite type, by characterizing algebraically the hyperbolic ends of G. 1. X = Poincar~ disc, G = conformaI automorphisms of X. 3. X = Cayley graph of a non-elementary finitely generated word hyperbolic group G.
Groups acting on hyperbolic Λ-Metric spaces
International Journal of Algebra and Computation, 2015
In this paper we study group actions on hyperbolic Λ-metric spaces, where Λ is an ordered abelian group. Λ-metric spaces were first introduced by Morgan and Shalen in their study of hyperbolic structures and then Chiswell, following Gromov's ideas, introduced the notion of hyperbolicity for such spaces. Only the case of 0-hyperbolic Λ-metric spaces (that is, Λ-trees) was systematically studied, while the theory of general hyperbolic Λ-metric spaces was not developed at all. Hence, one of the goals of the present paper was to fill this gap and translate basic notions and results from the theory of group actions on hyperbolic (in the usual sense) spaces to the case of Λ-metric spaces for an arbitrary Λ. The other goal was to show some principal difficulties which arise in this generalization and the ways to deal with them.
Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups
Advances in Mathematics, 2008
Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with infinite groups of outer automorphisms, and co-Hopfian relatively hyperbolic groups.
Hyperbolic Groups are Hyperhopfian
Journal of the Australian Mathematical Society, 2000
The main result indicates that every finitely generated, residually finite, torsion-free, cohopfian group having on free Abelian subgroup of rank two is hyperhopfian. The argument relies on earlier work and ideas of Hirshon. As a corollary, fundamental groups of all closed hyperbolic manifolds are hyperhopfian.
Relations Between Various Boundaries of Relatively Hyperbolic Groups
International Journal of Algebra and Computation, 2013
Suppose a group G is relatively hyperbolic with respect to a collection ℙ of its subgroups and also acts properly, cocompactly on a CAT(0) (or δ-hyperbolic) space X. The relatively hyperbolic structure provides a relative boundary ∂(G, ℙ). The CAT(0) structure provides a different boundary at infinity ∂X. In this paper, we examine the connection between these two spaces at infinity. In particular, we show that ∂(G, ℙ) is G-equivariantly homeomorphic to the space obtained from ∂X by identifying the peripheral limit points of the same type.
The Michigan Mathematical Journal, 1998
We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following.
Equivariant covers for hyperbolic groups
Geometry and Topology, 2008
We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture for K * (RG) for every word-hyperbolic group G and every coefficient ring R. 20F67; 37D40 We will construct the desired open covering U of G × X by pulling back V with the composition G × X j − → FS (X) φτ −→ FS (X) for an appropriate real number τ , where j is the map from Theorem 1.5. Obviously U has for every choice of τ all the desired properties except for the property that there exists U (g 0 ,c) ∈ U such that g α 0 × {c} ⊆ U (g 0 ,c) for every c ∈ X and every g 0 ∈ G. We conclude from Theorem 1.5 for τ ∈ R and the function f α appearing in Theorem 1.5 φ τ • j(g, c) ∈ φ [−β,β] (φ τ • j(g 0 , c)) fα(τ) for all c ∈ X and all g ∈ G with d G (g 0 , g) < α. By Theorem 1.5 there is τ such that f α (τ) < δ. For such a choice of τ we conclude from (1.7) that φ τ • j(g, c) ∈ φ [−β,β] (φ τ • j(g 0 , c)) δ ⊂ V φτ •j(g 0 ,c) for all c ∈ X and all g ∈ G with d G (g, g 0) < α. This finishes the proof of Theorem 1.2.
Endomorphisms of Relatively Hyperbolic Groups
International Journal of Algebra and Computation, 2008
We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out (G) is infinite, then G splits over a slender group. • If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an ℝ-tree is trivial, then H is Hopfian. • If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian. • Any finitely presented group is isomorphic to a finite index subgroup of Out (H) for some group H with Kazhdan property (T). (This sharpens a result of Ollivier–Wise).
Structure and Rigidity in (Gromov) Hyperbolic Groups and Discrete Groups in Rank 1 Lie Groups II
Geometric And Functional Analysis, 1997
We borrow the Jaco-Shalen-Johannson notion of characteristic submanifold from 3-dimensional topology to study cyclic splittings of torsion-free (Gromov) hyperbolic groups and finitely generated discrete groups in rank 1 Lie groups. Our JSJ canonical decomposition is a fundamental object for studying the dynamics of individual automorphisms and the automorphism group of a torsion-free hyperbolic group and a key tool in our approach to the isomorphism problem for these groups [S3]. For discrete groups in rank 1 Lie groups, the JSJ canonical decomposition serves as a basic object for understanding the geometry of the space of discrete faithful representations and allows a natural generalization of the Teichmüller modular group and the Riemann moduli space for these discrete groups.