Let $G$ be a connected simple linear Lie group of rank one, and let $\Gamma <G$ be a discrete Zariski dense subgroup admitting a finite Bowen-Margulis-Sullivan measure $m^{ \operatorname{ BMS}}$. We show that the right translation action of the one dimensional diagonalizable subgroup is mixing on $(\Gamma \backslash G, m^{ \operatorname{ BMS}})$. 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 $\Gamma \backslash G$, and provides precise asymptotics for the Haar measure matrix coefficients.
Categories: math.DS