Date of Award

Spring 2022

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Minsky, Yair

Abstract

The first part of this dissertation investigates the relationship between strata of Abelian differentials and various mapping class groups afforded by means of the topological monodromy representation. We show that the fundamental group of a stratum surjects onto the subgroup of the mapping class group which preserves a fixed framing of the underlying Riemann surface, thereby giving a complete characterization of the monodromy group. In the course of our proof we also show that these "framed mapping class groups" are finitely generated (even though they are of infinite index) and give explicit generating sets. This work was accomplished in collaboration with Nick Salter. The second part, representing joint work with James Farre, extends Mirzakhani's conjugacy between the earthquake and horocycle flows to a bijection. This yields conjugacies between these flows on all strata and exhibiting an abundance of new ergodic measures for the earthquake flow. The structure of our map indicates a natural extension of the earthquake flow to an action of the the upper-triangular subgroup P < SL2R and we classify the ergodic measures for this action as pullbacks of affine measures on the bundle of quadratic differentials. Our main tool in this part is a generalization of the shear coordinates of Bonahon and Thurston to arbitrary measured laminations.

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