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

Let $\mathrm{\Gamma }$ be a convex cocompact thin subgroup of an arithmetic lattice in $\mathrm{SO}(n,1)$. 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 $\mathrm{\Gamma }\mathrm{\backslash }{\mathbb{H}}^n$. 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 $\mathrm{\Gamma }$. These properties follow by proving that the return trajectory subgroups have finite index in $\mathrm{\Gamma }$.