Groups acting on hyperbolic Λ-Metric spaces (original) (raw)

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.

Hierarchy for groups acting on hyperbolic mathbfZn\mathbf{Z}^nmathbfZn-spaces

arXiv (Cornell University), 2016

In [8], the authors initiated a systematic study of hyperbolic Λmetric spaces, where Λ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case Λ = Z n taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic Z n-metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [18]) similar to the one established for Z n-free groups in [10].

A model for the universal space for proper actions of a hyperbolic group

2002

Let G be a word hyperbolic group in the sense of Gromov and P its associated Rips complex. We prove that the fixed point set P H is contractible for every finite subgroups H of G. This is the main ingredient for proving that P is a finite model for the universal space EG of proper actions. As a corollary we get that a hyperbolic group has only finitely many conjugacy classes of finite subgroups.

Symbolic dynamics and relatively hyperbolic groups

Groups, Geometry, and Dynamics, 2000

We study the action of a relatively hyperbolic group on its boundary, by methods of symbolic dynamics. Under a condition on the parabolic subgroups, we show that this dynamical system is finitely presented. We give examples where this condition is satisfied, including geometrically finite kleinian groups.

On metric relative hyperbolicity

2012

We show the equivalence of several characterizations of relative hyperbolicity for metric spaces, and obtain extra information about geodesics in a relatively hyperbolic space.

Notes on Hyperbolic and Automatic Groups

These notes are an expanded version of notes given by Panos Papasoglu, in an MSc course at the University of Warwick, 1994. I have always fancied expanding them further into a book but never quite got round to it. They also contain parts of my Warwick MSc Dissertation which studied Rips-Sela canonical representatives and the proof of Rips and Sela that there is an algorithm to determine whether a system of equations in a torsion-free hyperbolic group has a solution. I may expand them further in the future. Most likely I will move them in the direction of Bowditch's Cut Point Theorem and surrounding work, and the Dunwoody-Swenson generalisations of the JSJ decomposition and torus/annulus theorems, as this is probably the body of work which most impressed me when I was a PhD student.

Bounded geometry in relatively hyperbolic groups

If a group is relatively hyperbolic, the parabolic subgroups are virtually nilpotent if and only if there exists a hyperbolic space with bounded geometry on which it acts geometrically finitely. This provides, via the embedding theorem of M. Bonk and O. Schramm, a very short proof of the finiteness of asymptotic dimension for such groups (which is known to imply Novikov conjectures). Contents 1. Preliminaries 91 2. Polynomial growth for groups and bounded geometry for horoballs 92 3. Proof of Theorem 0.1 93 References 94

Towards the Definition of Metric Hyperbolicity

Moscow Mathematical Journal, 2005

We introduce measure-theoretic definitions of hyperbolic structure for measure-preserving automorphisms. A wide class of K-automorphisms possesses a hyperbolic structure; we prove that all K-automorphisms have a slightly weaker structure of semi-hyperbolicity. Instead of the notions of stable and unstable foliations and other notions from smooth theory, we use the tools of the theory of polymorphisms. The central role is played by polymorphisms associated with a special invariant equivalence relation, more exactly, with a homoclinic equivalence relation. We call an automorphism with given hyperbolic structure a hyperbolic automorphism and prove that it is canonically quasisimilar to a so-called prime nonmixing polymorphism. We present a short but necessary vocabulary of polymorphisms and Markov operators from [11, 12].