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

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

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