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

Abstract

Let $G:=\mathrm{SO}(n,1{)}^{\circ }$ and $\Gamma <G$ be a geometrically finite Zariski dense subgroup with critical exponent $\delta$ bigger than $(n-1)/2$. Under a spectral gap hypothesis on $L^2(\Gamma \backslash G)$, which is always satisfied when $\delta >(n-1)/2$ for $n=2,3$ and when $\delta >n-2$ for $n\ge 4$, we obtain an effective archimedean counting result for a discrete orbit of $\Gamma$ in a homogeneous space $H\backslash G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family $\{{\mathcal{B}}_T\subset H\backslash G\}$ of compact subsets, there exists $\eta >0$ such that

for an explicit measure $\mathcal{M}$ on $H\backslash G$ which depends on $\Gamma$. We also apply the 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 $L^2(\Gamma \backslash G)$ 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.