Mixing of frame flow for rank one locally symmetric spaces and measure classification

Abstract

Let G be a connected simple linear Lie group of rank one, and let Γ < G be a discrete Zariski dense subgroup admitting a finite Bowen-Margulis-Sullivan measure m BMS. We show that the right translation action of the one-dimensional diagonalizable subgroup is mixing on (Γ\G, m BMS). Together with the work of Roblin, this proves ergodicity of the Burger-Roblin measure under the horospherical group N, establishes a classification theorem for N invariant Radon measures on Γ\G, and provides precise asymptotics for the Haar measure matrix coefficients.