Symbolic dynamics and relatively hyperbolic groups

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. Associated to any word-hyperbolic group Γ, there is a dynamical system arising from the action of Γ on its Gromov boundary ∂Γ. Already in [12], M.Gromov uses methods of symbolic dynamics for the study of this action, and in [5] (see also [6]), M.Coornaert, and A.Papadopoulos explain a way to factorize such a dynamical system through a subshift of finite type. They describe a finite alphabetA, and a subshift Φ ⊂ AΓ, and they construct a continuous equivariant, surjective map Φ → ∂Γ, which encodes the action of Γ on its boundary by a subshift of finite type. The action of a group Γ on a compact metric space, K is expansive if there exists ε > 0 such that any pair of distinct points in K can be taken at distance at least ε from each other by an element of Γ. It is well known that the action of a hyperbolic group Γ on ∂Γ is expansive. This property, together with the existence of the coding given in [5], makes the action of a hyperbolic group, Γ, on its boundary, ∂Γ, finitely presented (see [12], [5]). In [12], M.Gromov describes consequences of such a presentation, like the rationality of some counting functions. The aim of this paper is to state and prove similar properties for relatively hyperbolic groups, where parabolic subgroups are allowed. In general, in presence of parabolics, the study of dynamical properties becomes significantly more complicated. After an idea of Gromov in [12], B.Farb [10] and B.Bowditch [2] developed the theory of relatively hyperbolic groups, as a generalization of geometrically finite Kleinian groups. We will use for this work the definition of relatively hyperbolic groups given by Bowditch in [2]. A group Γ is hyperbolic relative to a family, G, of finitely generated subgroups of Γ if it acts on an hyperbolic fine graph, with finite stabilizers of edges, finitely many orbits of edges, and such that the stabilizers of infinite valence vertices are exactly the elements of G (see Definition 2.3). In [10], this definition is equivalent to “relatively hyperbolic with the property BCP” (see the appendix of [9]). If one replaces “fine” by “locally finite” in above definition, then G is empty and the group is hyperbolic. In [10], Farb proves that the fundamental group of a finite volume manifold of pinched negative curvature, with finitely many cusps is hyperbolic relative to the conjugates of the fundamental groups of the cusps, which are virtually nilpotent. Sela’s limit groups, or, finitely generated ω-residually-free groups are hyperbolic relative to their maximal abelian non-cyclic subgroups, as shown in [8]. Bowditch describes a boundary for a relatively hyperbolic group in [2]. The group acts on this compact set as a convergence group, and the elements of the family G are parabolic subgroups for this action. Despite of those parabolic subgroups, the action is expansive (Proposition 3.18). Université Louis Pasteur, Strasbourg, e-mail : [email protected] University of Southampton, e-mail : [email protected]