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

Mohammadi, Amir, and Oh, Hee. "Matrix coefficients, counting and primes for orbits of geometrically finite groups." Journal of the European Mathematical Society 017.4 (2015): 837-897. <http://eudml.org/doc/277704>.

@article{Mohammadi2015,
abstract = {Let $G:=\mathrm \{SO\}(n,1)^\circ $ and $\Gamma (n-1)/2$ 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 $\lbrace \mathcal \{B\}_T\subset H \backslash G \rbrace $ 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.},
author = {Mohammadi, Amir, Oh, Hee},
journal = {Journal of the European Mathematical Society},
keywords = {geometrically finite group; matrix coefficients; mixing; sieve; spectral gap; equidistribution; geometrically finite group; matrix coefficients; mixing; sieve; spectral gap; equidistribution},
language = {eng},
number = {4},
pages = {837-897},
publisher = {European Mathematical Society Publishing House},
title = {Matrix coefficients, counting and primes for orbits of geometrically finite groups},
url = {http://eudml.org/doc/277704},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Mohammadi, Amir
AU - Oh, Hee
TI - Matrix coefficients, counting and primes for orbits of geometrically finite groups
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 4
SP - 837
EP - 897
AB - Let $G:=\mathrm {SO}(n,1)^\circ $ and $\Gamma (n-1)/2$ 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 $\lbrace \mathcal {B}_T\subset H \backslash G \rbrace $ 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.
LA - eng
KW - geometrically finite group; matrix coefficients; mixing; sieve; spectral gap; equidistribution; geometrically finite group; matrix coefficients; mixing; sieve; spectral gap; equidistribution
UR - http://eudml.org/doc/277704
ER -