Let and be a geometrically finite Zariski dense subgroup with critical exponent bigger than . Under a spectral gap hypothesis on , which is always satisfied when for and when for , we obtain an effective archimedean counting result for a discrete orbit of in a homogeneous space where is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family of compact subsets, there exists such that
for an explicit measure on which depends on . We also apply the affine sieve and describe the distribution of almost primes on orbits of in arithmetic settings.
One of key ingredients in our approach is an effective asymptotic formula for the matrix coefficients of 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.
Cite this article
Amir Mohammadi, Hee Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups. J. Eur. Math. Soc. 17 (2015), no. 4, pp. 837–897DOI 10.4171/JEMS/520