N ov 2 00 4 Bounded geometry in relatively hyperbolic groups

We prove that a group is hyperbolic relative to virtually nilpotent subgroups if and only if there exists a Gromov-hyperbolic metric space with bounded geometry on which it acts as a relatively hyperbolic group. As a consequence we obtain that any group hyperbolic relative to virtually nilpotent subgroups has finite asymptotic dimension. For these groups the Novikov conjecture holds. The class of relatively hyperbolic groups is an important class of groups encompassing hyperbolic groups, fundamental groups of geometrically finite orbifolds with pinched negative curvature, groups acting on CAT(0) spaces with isolated flats, and many other examples. It was introduced by M. Gromov in [G1] and developed by B. Bowditch, B. Farb, and other authors (eg: [Bow2],[F]). There is now an interesting and rich literature on the subject. A finitely generated group Γ is hyperbolic relative to a family of finitely generated subgroups G if it acts on a proper complete hyperbolic geodesic space X, preserving a family of disjoint horoballs {B p , p ∈ P }, finite up to the action of Γ, such that for all p, the stabiliser of B p is an element G p of G, that acts co-compactly on the horospheres of B p , and such that the action of Γ is co-compact on the complement of the horoballs (see [Bow2]). A space X satisfying the conditions of the definition is referred to as an associated space to Γ. Geometrically, one should think of the complement of the horoballs as of the cover of the convex core of a geometrically finite hyperbolic manifold (or equivalently of the thick part of the manifold for Margulis decomposition), and of the horoballs as of the covers of the cusps. In many geometrical examples, the parabolic subgroups of Γ, that is, the elements of the family G, are virtually nilpotent. The main examples are geometrically finite manifolds with pinched negative curvature (one can also mention limit groups [D2], groups with boundary homeomorphic to a Sierpinski curve or a 2-sphere [D3]). If the curvature is allowed to collapse to −∞, one can obtain other parabolic subgroups (especially non-amenable ones, see [GP] Prop.0.3). The difference between these two cases can be identified. Let us say that a space X is geometrically bounded if there exists a function f : R + → R + such that, for all R > 0, every ball of radius R can be covered …