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

Abstract

Let G := SO(n, 1)◦ and Γ (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 {BT ⊂ H\G} of compact subsets, there exists η > 0 such that #[e]Γ ∩ BT =M(BT ) +O(M(BT )) for an explicit measureM 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(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.