Exponential mixing of the geodesic flow on geometrically finite hyperbolic manifolds

We will discuss joint work with Hee Oh in which we establish exponential mixing of the geodesic flow for all geometrically finite hyperbolic (d+1)-manifolds with critical exponent greater than d/2. Key ingredients of our proof are the spectral gap of the Laplace operator on the manifold due to Lax-Phillips, and a careful study of the asymptotic behavior of matrix coefficients of complementary series representations of the special orthogonal groups SO(d+1,1).