Actions of fractional Dehn twists on moduli spaces

Summary

Abstract

We attempt to give a unified treatment of certain hyperbolic transformations, “fractional Dehn twists”, that induce “algebraic actions” on certain subspaces of moduli spaces.

This is done by considering the Teichmüller group Mod₀, {n} of the sphere with n unordered marked points and an action of this group on M₀, n the moduli space of the sphere with n ordered marked points. We show that for covers of the sphere ramified over n points the above action lifts to the action of fractional Dehn twists.

Introduction

This paper is an attempt to give a unified treatment of scattered results accumulated over the years on “algebraic actions” of fractional Dehn twists, where by algebraic actions we mean that they induce algebraic transformations on moduli.

The first evidence of the existence of such actions is probably in [BuSi] where it was in fact not realized that they were fractional Dehn twists. They were only identified as such later by A. Aigon in [Ai].

Other examples, in genus 2 and 3, were found in [AiSi] and [Si1]. All these examples have in common the fact that they concerned families of surfaces with non-trivial automorphisms, either hyperelliptic surfaces with an extra non-hyperelliptic automorphism or non-hyperelliptic surfaces with a non-trivial automorphism.