Exponential mixing of frame flows for convex cocompact locally symmetric spaces

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 $G=\mathrm{SO}(n,1{)}^{\circ }$ and also strengthens the mixing result of Winter in the convex cocompact case.

28 pages