Matrix coefficients, Counting and Primes for orbits of geometrically finite groups

Abstract: Let G:=SO(n,1)^\circ and \Gamma be a geometrically finite Zariski dense subgroup with critical exponent delta bigger than (n-1)/2. Under a spectral gap hypothesis on L^2(\Gamma \ G), which is always satisfied for delta>(n-1)/2 for n=2,3 and for delta>n-2 for n>= 4, we obtain an {\it effective} archimedean counting result for a discrete orbit of \Gamma in a homogeneous space H\G, where H is the trivial group, an affine symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family {B_T} of compact subsets in H\G, there exists \eta>0 such that #[e]\G\cap B_T=M(B_T) +O(M(B_T)^{1-\eta}) for an explicit measure M on H\G, which depends on Gamma. We also apply affine sieve and describe the distribution of almost primes on orbits of \Gamma 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. An effective mixing for the Bowen-Margulis-Sullivan measure is also obtained as an application of our methods.