Roots of Dehn twists about multicurves

TL;DR

This paper characterizes when roots of Dehn twists about multicurves exist on surfaces, providing combinatorial criteria, bounds on root orders, and classifying all roots for genus 3 and 4 surfaces.

Contribution

It establishes necessary and sufficient conditions for roots of multicurve Dehn twists, introduces combinatorial data for roots, and classifies all such roots for genus 3 and 4 surfaces.

Findings

Necessary and sufficient conditions for roots of Dehn twists

Combinatorial data characterizing roots

Classification of roots for genus 3 and 4 surfaces

Abstract

A \textit{multicurve} $\C$ on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist $t_{\C}$ about $\C$ is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the existence of a root of such a Dehn twist, that is, a homeomorphism $h$ such that $h^n = t_{\C}$. We give combinatorial data that corresponds to such roots, and use it to determine upper bounds for $n$. Finally, we classify all such roots up to conjugacy for surfaces of genus 3 and 4.