Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems

Open questions 100 1 Appendix. Equivalent definitions of relative hyperbolicity 103 Bibliography 108 2 Chapter 1 Introduction 1.1 Preliminary remarks Originally, the notion of a relatively hyperbolic group was proposed by Gromov [45] in order to generalize various examples of algebraic and geometric nature such as fundamental groups of finite–volume non–compact Riemannian mani-folds of pinched negative curvature, geometrically finite Kleinian groups, word hyperbolic groups, small cancellation quotients of free products, etc. Gromov's idea has been elaborated by Bowditch in [13]. (An alternative approach was suggested by Farb [37].) In the present paper we obtain a characterization of relative hyperbolicity in terms of isoperimetric inequalities and adopt techniques based on van Kampen diagrams to the study of algebraic and algorithmic properties of relatively hyperbolic groups. This allows to establish a background for the subsequent paper [68], where we use relative hyperbolicity to prove embedding theorems for countable groups. Since the words 'relatively hyperbolic group' seem to mean different things for different people, we briefly explain here our terminology. There are two different approaches to the definition of the relative hyperbolicity of a group G with respect to a collection of subgroups {H 1 ,. .. , H m }. The first one was suggested by Bowditch [13]. It is similar to the original Gromov's concept and characterizes relative hyperbolicity in terms of the dynamics of properly discon-tinuous isometric group actions on hyperbolic spaces. (For exact definitions we refer to the appendix). In the paper [37], Farb formulated another definition in terms of the coset graphs. In the simplest case of a group G generated by a finite set S and one subgroup H ≤ G it can be stated as follows. G is hyperbolic relative to H if the graph Γ(G, S) obtained from the Cayley graph Γ(G, S) of G by contracting each of the cosets gH, g ∈ G, to a point is hyperbolic. In fact, the hyperbolicity of Γ(G, S) is independent on the choice of the finite generating set S in G. The two definitions were compared in [80], where Szczepa´nski showed that if 3 a group G is hyperbolic with respect to a collection of subgroups {H 1 ,. .. , H m } in the sense of Bowditch, then G is hyperbolic with respect to {H 1 ,. .. , H m } in the sense of Farb, but not conversely. However, in [37] Farb …