The Realization Problem for Dilation Surfaces

TL;DR

This paper investigates which elements of the mapping class group can be realized as affine automorphisms of dilation surfaces, revealing the existence of exotic Dehn twists and providing a construction method for such surfaces.

Contribution

It characterizes the types of mapping class group elements that can be realized as affine automorphisms of dilation surfaces and generalizes Thurston's construction to this setting.

Findings

Dilation surfaces can have exotic Dehn twists in their automorphism groups.

Only certain mapping class group elements can be realized as affine automorphisms.

A generalized Thurston construction produces dilation surfaces realizing pairs of Dehn multitwists.

Abstract

Dilation surfaces, or twisted quadratic differentials, are variants of translation surfaces. In this paper, we study the question of what elements or subgroups of the mapping class group can be realized as affine automorphisms of dilation surfaces. We show that dilation surfaces can have exotic Dehn twists in their affine automorphism groups and will establish that only certain types of mapping class group elements can arise as affine automorphisms of dilation surfaces. We also generalize a construction of Thurston that constructs a translation surface from a pair of filling multicurves to dilation surfaces. This construction will give us dilation surfaces that realize a pair of Dehn multitwists in their affine automorphism groups.