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

Abstract

Let Γ be a geometrically finite discrete subgroup in SO (d+ 1 , 1) ∘ with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle T 1(Γ \ Hd+1) with respect to the Bowen–Margulis–Sullivan measure, which is the unique probability measure on T 1(Γ \ Hd+1) with maximal entropy. As an application, we obtain a resonance-free region for the resolvent of the Laplacian on Γ \ 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.