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

Let G := SO(n, 1)degrees and Gamma < G be a geometrically finite Zariski dense subgroup with critical exponent delta greater than (n - 1)/2. Under a spectral gap hypothesis on L-2(Gamma\G), which is always satisfied when delta > (n - 1)/2 for n = 2,3 and when delta > n - 2 for n >= 4, we obtain an effective archimedean counting result for a discrete orbit of Gamma 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 subset of H\G} of compact subsets, there exists eta > 0 such that #[e]Gamma boolean AND B-T = m(B-T) + O(M(B-T)(1-eta)) for an explicit measure M on H\G which depends on F. We also apply the affine sieve and describe the distribution of almost primes on orbits of F in arithmetic settings. One of key ingredients in our approach is an effective asymptotic formula for the matrix coefficients of L-2(Gamma\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.