Matrix coefficients, counting and primes for orbits of geometrically finite groups Article Swipe
YOU?
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· 2015
· Open Access
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· DOI: https://doi.org/10.4171/jems/520
· OA: W2167696231
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 \#[e]\Gamma\cap \mathcal B_T=\mathcal M(\mathcal B_T) +O(\mathcal M(\mathcal B_T)^{1-\eta}) 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.
