# 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 $G:=SO(n,1)_{∘}$ and $Γ<G$ be a geometrically finite Zariski dense subgroup with critical exponent $δ$ bigger than $(n−1)/2$. Under a spectral gap hypothesis on $L_{2}(Γ\G)$, 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 $Γ$ 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 ${B_{T}⊂H\G}$ of compact subsets, there exists $η>0$ such that

for an explicit measure $M$ 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 $L_{2}(Γ\G)$ 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.

## 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