A finiteness result for groups which quasi-act on hyperbolic spaces

Let (X, d) be a Gromov-hyperbolic metric space endowed with a measure having finite entropy H and such that the measure of every ball of radius R > 0 is finite and bounded from below by a positive function of R. In this paper we look at the set Q(X; L, C, D) of the isomorphism classes of torsion-free groups Γ which admit a discrete, D-co-bounded (L, C)-quasi-action on X (D > 0, L ≥ 1, C ≥ 0) and we describe some algebraic conditions which, imposed on the groups Γ, define finite subsets of Q(X; L, C, D), provided C < ε for some ε > 0. As an example, these conditions are satisfied when Γ is assumed to admit a faithful, discrete, m-dimensional representation over some local field (in this case ε = ε(m, H, L)). In particular (set C = 0, L = 1), our results apply when the groups are assumed to act by isometries.