Roots of Dehn twists about separating curves

TL;DR

This paper classifies roots of Dehn twists about separating curves on surfaces, using compatible actions called nestled actions, and provides bounds on their degrees for genus 2 and 3 surfaces.

Contribution

It introduces a classification method for roots of Dehn twists via data sets and compatible actions, and establishes bounds on root degrees for low-genus surfaces.

Findings

All roots are derived from pairs of nestled actions satisfying a compatibility condition.

Classified all roots for genus 2 and 3 surfaces.

Established lower and upper bounds for the degrees of roots.

Abstract

Let $C$ be a curve in a closed orientable surface $F$ of genus $g \geq 2$ that separates $F$ into subsurfaces $\widetilde {F_i}$ of genera $g_i$, for $i = 1,2$. We study the set of roots in $\Mod(F)$ of the Dehn twist $t_C$ about $C$. All roots arise from pairs of $C_{n_i}$-actions on the $\widetilde{F_i}$, where $n=\lcm(n_1,n_2)$ is the degree of the root, that satisfy a certain compatibility condition. The $C_{n_i}$ actions are of a kind that we call nestled actions, and we classify them using tuples that we call data sets. The compatibility condition can be expressed by a simple formula, allowing a classification of all roots of $t_C$ by compatible pairs of data sets. We use these data set pairs to classify all roots for $g = 2$ and $g = 3$. We show that there is always a root of degree at least $2g^2+2g$, while $n \leq 4g^2+2g$. We also give some additional applications.