Anosov groups: local mixing, counting and equidistribution

Edwards, Samuel; Lee, Minju; Oh, Hee

doi:10.2140/gt.2023.27.513

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Anosov groups: local mixing, counting and equidistribution

Samuel Edwards, Minju Lee and Hee Oh

Geometry & Topology 27 (2023) 513–573

DOI: 10.2140/gt.2023.27.513

Abstract

Let $G$ be a connected semisimple real algebraic group, and $Γ < G$ a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients $⟨ (\exp t v) . f_{1}, f_{2} ⟩$ in $L^{2} (Γ ∖ G)$ as $t \to \infty$ for any $f_{1}, f_{2} \in C_{c} (Γ ∖ G)$ and any vector $v$ in the interior of the limit cone of $Γ$ . These asymptotics involve higher-rank analogues of Burger–Roblin measures, which are introduced in this paper. As an application, for any affine symmetric subgroup $H$ of $G$ , we obtain a bisector counting result for $Γ$ –orbits with respect to the corresponding generalized Cartan decomposition of $G$ . Moreover, we obtain analogues of the results of Duke, Rudnick and Sarnak as well as Eskin and McMullen for counting discrete $Γ$ –orbits in affine symmetric spaces $H ∖ G$ .

Keywords

Anosov group, local mixing, counting, equidistribution, higher rank Patterson–Sullivan theory

Mathematical Subject Classification

Primary: 22E40, 37A17, 37A25, 37A44

References

Publication

Received: 12 May 2020

Revised: 15 September 2021

Accepted: 3 November 2021

Published: 16 May 2023

Proposed: Anna Wienhard

Seconded: David M Fisher, Dmitri Burago

Authors

Samuel Edwards
Department of Mathematics Yale University New Haven, CT United States	Department of Mathematics Durham University Durham United Kingdom

Minju Lee
Department of Mathematics Yale University New Haven, CT United States	Department of Mathematics University of Chicago Chicago, IL United States

Hee Oh
Department of Mathematics Yale University New Haven, CT United States	Korea Institute for Advanced Study Seoul South Korea

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Volume 27, issue 2 (2023)