Limit sets of discrete groups of isometries of exotic hyperbolic spaces

Abstract

Let Γ \Gamma be a geometrically finite discrete group of isometries of hyperbolic space H F n \mathcal {H}_{\mathbb {F}}^n , where F = R , C , H \mathbb {F}= \mathbb {R}, \mathbb {C}, \mathbb {H} or O \mathbb {O} (in which case n = 2 n=2 ). We prove that the critical exponent of Γ \Gamma equals the Hausdorff dimension of the limit sets Λ ( Γ ) \Lambda (\Gamma ) and that the smallest eigenvalue of the Laplacian acting on square integrable functions is a quadratic function of either of them (when they are sufficiently large). A generalization of Hopf ergodicity theorem for the geodesic flow with respect to the Bowen-Margulis measure is also proven.