Anosov groups: local mixing, counting and equidistribution

Edwards, Samuel¹ ; Lee, Minju² ; Oh, Hee³

¹ Department of Mathematics, Yale University, New Haven, CT, United States, Department of Mathematics, Durham University, Durham, United Kingdom
² Department of Mathematics, Yale University, New Haven, CT, United States, Department of Mathematics, University of Chicago, Chicago, IL, United States
³ Department of Mathematics, Yale University, New Haven, CT, United States, Korea Institute for Advanced Study, Seoul, South Korea

Geometry & topology, Tome 27 (2023) no. 2, pp. 513-573.

Résumé

Let $G$ be a connected semisimple real algebraic group, and $\Gamma <G$ a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients $\langle (\exp tv).f_1,f_2\rangle$ in $L^2(\Gamma \setminus G)$ as $t\to \infty$ for any $f_1,f_2\in C_c(\Gamma \setminus G)$ and any vector $v$ in the interior of the limit cone of $\Gamma$. 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 $\Gamma$–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 $\Gamma$–orbits in affine symmetric spaces $H\setminus G$.

Détail

DOI : 10.2140/gt.2023.27.513

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

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