  Let $\Gamma$ 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
$\operatorname{T}^1(\Gamma\backslash \mathbb{H}^{d+1})$ with respect to the
Bowen-Margulis-Sullivan measure, which is the unique probability measure on
$\operatorname{T}^1(\Gamma\backslash \mathbb{H}^{d+1})$ with maximal entropy.
As an application, we obtain a resonance free region for the resolvent of the
Laplacian on $\Gamma\backslash \mathbb{H}^{d+1}$. Our approach is to construct
a coding for the geodesic flow and then prove a Dolgopyat-type spectral
estimate for the corresponding transfer operator.
