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

### Amir Mohammadi

The University of Texas at Austin, USA### Hee Oh

Yale University, New Haven, USA

## 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 $n\geq 4$, 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.

## 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. 837–897

DOI 10.4171/JEMS/520