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


Let G:=SO(n,1)G:=\mathrm {SO}(n,1)^\circ and Γ<G\Gamma<G be a geometrically finite Zariski dense subgroup with critical exponent δ\delta bigger than (n1)/2(n-1)/2. Under a spectral gap hypothesis on L2(Γ\G)L^2(\Gamma \backslash G), which is always satisfied when δ>(n1)/2\delta>(n-1)/2 for n=2,3n=2,3 and when δ>n2\delta>n-2 for n4n\geq 4, we obtain an effective archimedean counting result for a discrete orbit of Γ\Gamma in a homogeneous space H\GH \backslash G where HH is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family {BTH\G}\{\mathcal B_T\subset H \backslash G \} of compact subsets, there exists η>0\eta>0 such that

#[e]ΓBT=M(BT)+O(M(BT)1η)\#[e]\Gamma\cap \mathcal B_T=\mathcal M(\mathcal B_T) +O(\mathcal M(\mathcal B_T)^{1-\eta})

for an explicit measure M\mathcal M on H\GH\backslash G which depends on Γ\Gamma. We also apply the affine sieve and describe the distribution of almost primes on orbits of Γ\Gamma in arithmetic settings.

One of key ingredients in our approach is an effective asymptotic formula for the matrix coefficients of L2(Γ\G)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