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

Abstract

Let Γ be a convex cocompact thin subgroup of an arithmetic lattice in SO(n,1). We generalize Selberg's 316 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 Γ﹨Hn. 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 Γ. These properties follow by proving that the return trajectory subgroups are of finite index in Γ.