Dehn filling in relatively hyperbolic groups (original) (raw)
Related papers
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.
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
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.
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.
Finite groups and hyperbolic manifolds
Inventiones mathematicae, 2005
The isometry group of a compact n-dimensional hyperbolic manifold is known to be finite. We show that for every n ≥ 2, every finite group is realized as the full isometry group of some compact hyperbolic n-manifold. The cases n = 2 and n = 3 have been proven by Greenberg [G] and Kojima [K], respectively. Our proof is non constructive: it uses counting results from subgroup growth theory to show that such manifolds exist.
Quasi-hyperbolic planes in relatively hyperbolic groups
Arxiv preprint arXiv:1111.2499, 2011
We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit certain splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. The specific embeddings we find remain quasiisometric embeddings when composed with the natural map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to "almost every" peripheral (Dehn) filling.
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.
Some Examples of Hyperbolic Groups
… theory down under īCanberra, 2006
We describe some examples of hyperbolic groups to compare the range of validity of different theories of JSJ-decompositions of groups.
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.