The geometry and dynamics of twisted quadratic differentials

Abstract

This thesis examines twisted quadratic differentials, also known as dilation surfaces. These are variants of translation surfaces, their more well-studied counterpart. In this work, we study questions about the realizability of mapping class group elements in the affine automorphism groups of dilation surfaces, and how large affine automorphism groups can be. We demonstrate how to construct dilation surfaces with a given pseudo-Anosov map in their affine automorphism group, show the existence of exotic Dehn twists, and construct dilation surfaces with simultaneous Dehn multitwists. The last construction also gives rise to some large affine automorphism groups.