Matrix coefficients, counting and primes for orbits of geometrically finite groups

Let G:=SO(n,1)^∘ and Γ be a geometrically finite Zariski dense subgroup with critical exponent δ bigger than (n-1)/2 . Under a spectral gap hypothesis on L^2(Γ G) , which is always satisfied when δ>(n-1)/2 for n=2,3 and when δ>n-2 for n≥ 4 , we obtain an effective archimedean counting result for a discrete orbit of Γ in a homogeneous space H G where H is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family \{ B_T⊂ H G \} of compact subsets, there exists η>0 such that #[e]Γ∩ B_T= M( B_T) +O( M( B_T)1-η) for an explicit measure M on H G which depends on Γ . We also apply the affine sieve and describe the distribution of almost primes on orbits of Γ in arithmetic settings. One of key ingredients in our approach is an effective asymptotic formula for the matrix coefficients of L^2(Γ G) that we prove by combining methods from spectral analysis, harmonic analysis and ergodic theory. We also prove exponential mixing of the frame flows with respect to the Bowen-Margulis-Sullivan measure.