Spectral gap and exponential mixing on geometrically finite hyperbolic manifolds

Let M=Γ∖Hd+1 be a geometrically finite hyperbolic manifold with critical exponent exceeding d∕2. We obtain precise asymptotic expansion of the matrix coefficients for geodesic flow in L2(T1(M)), exponential error term essentially as good one given by spectral gap Laplace operator on L2(M) due to Lax and Phillips. Combined work Bourgain, Gamburd, Sarnak its generalization Golsefidy Varjú expanders, this implies uniform mixing congruence covers M when Γ is Zariski-dense subgroup contained an arithmetic SO∘(d,1).