Fractional powers of Dehn twists about nonseparating curves

Abstract

Let $S_g$ be a closed orientable surface of genus $g\ge 2$ 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 $n\le 2g+1$. In general, we show that $n\le 4g$, 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 $2n\ge 2g+2$, 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 $S_5$.

arXiv comment16 pages, 1 figure