Generalization of Selberg's $3/16$ theorem for convex cocompact thin subgroups of $\operatorname{SO}(n, 1)$ (2006.07787v2)

Abstract: Let $\Gamma$ be a convex cocompact thin subgroup of an arithmetic lattice in $\operatorname{SO}(n, 1)$. We generalize Selberg's $\frac{3}{16}$ theorem in this setting, i.e., we prove uniform exponential mixing of the frame flow and obtain a uniform resonance-free half plane for the congruence covers of the hyperbolic manifold $\Gamma \backslash \mathbb H^n$. This extends the work of Oh-Winter who established the $n = 2$ case. The theorem follows from uniform spectral bounds for the congruence transfer operators with holonomy. We employ Sarkar-Winter's frame flow version of Dolgopyat's method uniformly over the congruence covers as well as Golsefidy-Varj\'{u}'s generalization of Bourgain-Gamburd-Sarnak's expansion machinery by using the properties that the return trajectory subgroups are Zariski dense and have trace fields which coincide with that of $\Gamma$. These properties follow by proving that the return trajectory subgroups have finite index in $\Gamma$.