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Anosov groups: local mixing, counting and equidistribution

Samuel Edwards, Minju Lee and Hee Oh

Geometry & Topology 27 (2023) 513–573
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 tv).f1,f2 in L2(ΓG) as t for any f1,f2 Cc(ΓG) 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 HG.

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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