Fractional powers of Dehn twists about nonseparating curves

TL;DR

This paper characterizes fractional powers of Dehn twists about nonseparating curves on surfaces, establishing existence conditions, bounds on exponents, and explicit examples, enriching the understanding of mapping class group elements.

Contribution

It provides necessary and sufficient conditions for the existence of fractional powers, including side-exchanging and side-preserving cases, with explicit bounds and examples.

Findings

Side-preserving fractional powers exist with n ≤ 2g+1.

Side-exchanging fractional powers exist with specific exponents, e.g., (2g+2)/(4g+2).

Complete classification of certain fractional powers on S_5.

Abstract

Let $S_g$ be a closed orientable surface of genus $g \geq 2$ and $C$ a simple closed nonseparating curve in $F$. Let $t_C$ denote a left handed Dehn twist about $C$. A \textit{fractional power} of $t_C$ of \textit{exponent} $\fraction{\ell}{n}$ is an $h \in \Mod(S_g)$ such that $h^n = t_C^{\ell}$. Unlike a root of a $t_C$, a fractional power $h$ can exchange the sides of $C$. We derive necessary and sufficient conditions for the existence of both side-exchanging and side-preserving fractional powers. We show in the side-preserving case that if $\gcd(\ell,n) = 1$, then $h$ will be isotopic to the $\ell^{th}$ power of an $n^{th}$ root of $t_C$ and that $n \leq 2g+1$. In general, we show that $n \leq 4g$, and that side-preserving fractional powers of exponents $\fraction{2g}{2g+2}$ and $\fraction{2g}{4g}$ always exist. For a side-exchanging fractional power of exponent $\fraction{\ell}{2n}$, we show that $2n \geq 2g+2$, and that side-exchanging fractional powers of exponent $\fraction{2g+2}{4g+2}$ and $\fraction{4g+1}{4g+2}$ always exist. We give a complete listing of certain side-preserving and side-exchanging fractional powers on $S_5$.