Anosov groups: local mixing, counting and equidistribution

Edwards, Samuel; Lee, Minju; Oh, Hee

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 ⟨(exptv). f1,f2⟩ in L2(Γ∖G) as t→∞ for any f1,f2Cc(Γ∖G) and any vector v in them 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 Hof 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.

Citation

Edwards, S., Lee, M., & Oh, H. (2023). Anosov groups: local mixing, counting and equidistribution. Geometry & Topology, 27(2), 513-573. https://doi.org/10.2140/gt.2023.27.513

• Journal Article Type: Article
• Acceptance Date: Nov 3, 2021
• Online Publication Date: May 16, 2023
• Publication Date: May 16, 2023
• Deposit Date: Feb 19, 2024
• Publicly Available Date: Feb 19, 2024
• Journal: Geometry & Topology
• Print ISSN: 1465-3060
• Electronic ISSN: 1465-3060
• Publisher: Mathematical Sciences Publishers (MSP)
• Peer Reviewed: Peer Reviewed
• Volume: 27
• Issue: 2
• Pages: 513-573
• DOI: https://doi.org/10.2140/gt.2023.27.513
• Keywords: Geometry and Topology
• Public URL: https://durham-repository.worktribe.com/output/2269548