Effective equidistribution of horospherical flows in infinite volume rank one homogeneous spaces

Abstract

We prove effective equidistribution of horospherical flows in $\operatorname{SO}(n,1)^\circ / \Gamma$ when $\Gamma$ is geometrically finite and the frame flow is exponentially mixing for the Bowen-Margulis-Sullivan measure. We also discuss settings in which such an exponential mixing result is known to hold. As part of the proof, we show that the Patterson-Sullivan measure satisfies some friendly-like properties when $\Gamma$ is geometrically finite.