Exponential mixing of frame flows for convex cocompact locally symmetric spaces

TL;DR

This paper proves that the frame flow on certain convex cocompact locally symmetric spaces exhibits exponential mixing, extending previous results and employing generalized non-integrability and non-concentration conditions.

Contribution

It generalizes exponential mixing results for frame flows to a broader class of rank one groups and convex cocompact subgroups, using new analytical conditions.

Findings

Proves exponential mixing of frame flows for convex cocompact spaces

Generalizes previous results from specific groups to broader classes

Introduces generalized non-integrability and non-concentration conditions

Abstract

Let $G$ be a connected center-free simple real algebraic group of rank one and $\Gamma < G$ be a Zariski dense torsion-free convex cocompact subgroup. We prove that the frame flow on $\Gamma \backslash G$, i.e., the right translation action of a one-parameter subgroup $\{a_t\}_{t \in \mathbb R} < G$ of semisimple elements, is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The key step is proving suitable generalizations of the local non-integrability condition and the non-concentration property which are essential for Dolgopyat's method. This generalizes the work of Sarkar-Winter for $G = \operatorname{SO}(n, 1)^\circ$ and also strengthens the mixing result of Winter in the convex cocompact case.