CHAPTER 1

Introduction

1.1. Preliminary remarks

Originally, the notion of a relatively hyperbolic group was proposed by Gro-

mov [45] in order to generalize various examples of algebraic and geometric nature

such as fundamental groups of finite-volume non-compact Riemannian manifolds

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 hyper-

bolic groups. This allows us to establish a background for the subsequent paper

[72], where we use relative hyperbolicity to prove embedding theorems for count-

able groups and to construct groups with certain 'exotic' properties. For instance,

in [72] we construct first examples of finitely generated groups other than Z/2Z

with exactly 2 conjugacy classes.

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 re-

spect to a collection of subgroups {Hi,..., #

m

} . The first one was suggested by

Bowditch [13]. It is similar to Gromov's original concept and characterizes relative

hyperbolicity in terms of the dynamics of properly discontinuous 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 r(G, S) obtained from the Cayley graph T(G, S) of G by contracting each of

the cosets gH, g G G, to a point is hyperbolic. In fact, the hyperbolicity of T(G, 5)

is independent of the choice of the finite generating set S in G.

The two definitions were compared in [84], where Szczepahski showed that if

a group G is hyperbolic with respect to a collection of subgroups {Hi,..., Hm}

in the sense of Bowditch, then G is hyperbolic with respect to {Hi,... ,Hm} in

the sense of Farb, but not conversely. However, in [37] Farb does not simply

consider relatively hyperbolic groups. He introduces an additional (and central in

his theory) condition, the so-called Bounded Coset Penetration property (or BCP,

for brevity). It turns out that the notion of relative hyperbolicity with BCP in the

sense of Farb is equivalent to the notion of relative hyperbolicity in the sense of

Bowditch [21, 32]. In what follows, to distinguish these two notions we use words

'weakly relatively hyperbolic' (respectively 'relatively hyperbolic') for groups that

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