UC San Diego

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 a significant part of the proof, we establish quantitative nondivergence of horospherical orbits, and show that the Patterson-Sullivan measure is more than friendly when $\Gamma$ is geometrically finite. We also prove that a much stronger result, called global friendliness, if all cusps are assumed to be of maximal rank. The proof strategy of the equidistribution theorems combines these with the ``banana trick'' of Margulis. As an application, we study the distribution of non-discrete orbits of geometrically finite groups in $\SO(n,1)$ acting on the quotient of $\operatorname{SO}(n,1)$ by a horospherical subgroup. In particular, this can be identified with $\Gamma$ acting on the ``light cone'' in $\mathbb{R}^{n+1}$, or on certain wedge products. We obtain asymptotics for the distribution of orbits of general geometrically finite groups. When the Bowen-Margulis-Sullivan measure is exponentially mixing and all cusps have maximal rank, we obtain a quantitative ratio theorem, using global friendliness of the PS measure.