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

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 4, page 837-897
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topMohammadi, 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.