Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds

Let $\mathcal{M}=\Gamma\backslash\mathbb{H}^{d+1}$ be a geometrically finite hyperbolic manifold with critical exponent exceeding $d/2$. We obtain a precise asymptotic expansion of the matrix coefficients for the geodesic flow in $L^2(\mathrm{T}^1(\mathcal{M}))$, with exponential error term essentially as good as the one given by the spectral gap for the Laplace operator on $L^2(\mathcal{M})$ due to Lax and Phillips. Combined with the work of Bourgain, Gamburd, and Sarnak and its generalization by Golsefidy and Varju on expanders, this implies uniform exponential mixing for congruence covers of $\mathcal{M}$ when $\Gamma$ is a thin subgroup of $\mathrm{SO}^{\circ}(d+1,1)$. Our result implies that, with respect to the Bowen-Margulis-Sullivan measure, the geodesic flow on $\mathrm{T}^1(\mathcal{M})$ is exponentially mixing, uniformly over congruence covers in the case when $\Gamma$ is a thin subgroup.