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

Abstract

Let $\Gamma$ 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 $\Gamma \backslash {\mathbb{H}}^n$. 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-Varjú'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 $\Gamma$.

This paper supersedes the geodesic flow version arXiv:1903.00825. The second version includes a new theorem that the return trajectory subgroups are of finite index. 51 pages, 2 figures. arXiv admin note: text overlap with arXiv:2004.14551

References in corpus (2)

• Wave decay on convex co-compact hyperbolic manifolds
• Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

Cited by in corpus (1)

• The fractal uncertainty principle via Dolgopyat's method in higher dimensions