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

Abstract

Let G := SO(n, 1)° and F 2 (F\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 F 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 {BT ⊂ H \G} of compact subsets, there exists n > 0 such that #[e]F ∩ BT = M(BT ) + O(M(B T) 1-n) 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 (F\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.