Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

Let Gamma be a geometrically finite discrete subgroup in SO(d + 1, 1)degrees with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle T-1 (Gamma\Hd+1) with respect to the Bowen-Margulis- Sullivan measure, which is the unique probability measure on T-1 (Gamma\Hd+1) with maximal entropy. As an application, we obtain a resonance-free region for the resolvent of the Laplacian on Gamma\Hd+1. Our approach is to construct a coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator.