Critical Exponents for Groups of Isometries

Let Γ be a convex co-compact group of isometries of a CAT(−1) space X and let Γ0 be a normal subgroup of Γ. We show that, provided Γ is a free group, a sufficient condition for Γ and Γ0 to have the same critical exponent is that Γ/Γ0 is amenable. 0. Introduction and Results Let Γ be a group of isometries acting freely and properly discontinuously on a CAT(−1) space X . Roughly speaking, a CAT(−1) space is a path metric space for which every geodesic triangle is more pinched than a congruent triangle in the hyperbolic plane; see [5] for a formal definition. Prototypical examples of CAT(−1) spaces are simply connected Riemannian manifold with sectional curvatures bounded above by −1 and (simplicial or non-simplicial) R-trees. A fundamental quantity associated to Γ is its critical exponent δ(Γ). This is defined to be the abscissa of convergence of the Poincaré series