Exponential mixing of frame flows for geometrically finite hyperbolic manifolds

Abstract

We establish that frame flows for geometrically finite hyperbolic manifolds of arbitrary dimensions $\Gamma \backslash {\mathbb{H}}^{d+1}$ are exponentially mixing with respect to the Bowen–Margulis–Sullivan measure, which is the measure of maximal entropy. This paper focuses on the remaining case, the case with cusps. To prove this, we utilize the countably infinite symbolic coding and perform a frame flow version of Dolgopyat’s method à la Sarkar–Winter and Tsujii–Zhang. This requires the local non-integrability condition and the non-concentration property but the challenge in the presence of cusps is that the latter holds only on a large proper subset. To overcome this, we use an effective renewal theorem to prove a uniform large deviation property for symbolic recurrence to the large subset, inspired by the work of Li. Applications of the main theorem include an asymptotic formula for matrix coefficients for $L^2(\Gamma \backslash \mathrm{SO}(d+1,1{)}^{\circ })$ with an exponential error term, and exponential equidistribution of holonomies and translates of horospherical orbits.