Date of Award

Spring 2022

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Oh, Hee

Abstract

The main objective of this thesis is to prove uniform exponential mixing of the frame flow for congruence covers of a hyperbolic manifold $\Gamma \backslash \mathbb H^n$ with respect to the Bowen--Margulis--Sullivan measures, for some convex cocompact thin subgroup $\Gamma$ of an arithmetic lattice in $\SO(n, 1)$. In order to accomplish this, we first need to prove exponential mixing of the frame flow for a single convex cocompact hyperbolic manifold $\Gamma \backslash \mathbb H^n$ as well. The former result extends the work of Oh--Winter who established the $n = 2$ case. The theorems follow from (uniform) spectral bounds for the (congruence) transfer operators with holonomy. For a single convex cocompact hyperbolic manifold, we develop a frame flow version of Dolgopyat's method. This requires proving the key properties called the local non-integrability condition and the non-concentration property. For congruence covers of a convex cocompact hyperbolic manifold, we do the same uniformly over the congruence covers. However, in this case, we also need to use Golsefidy--Varj\'{u}'s generalization of Bourgain--Gamburd--Sarnak's expander machinery. This requires introducing the new concept of return trajectory subgroups and proving the key properties that they are Zariski dense and have trace fields which coincide with that of $\Gamma$. Immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. We also give applications for which the uniformity is essential. The first is an affine sieve and the second is a uniform resonance-free half plane for the resolvents of the Laplacians. Lastly, we also present results from two other highly related works. The first is regarding a uniform asymptotic orbit counting formula for congruence subsemigroups of $\SO(n, 1)$. The second is regarding local and topological mixing of one-parameter diagonal flows on Anosov homogeneous spaces which are the higher rank analogue of frame flows for convex cocompact hyperbolic manifolds.

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