Roots of Dehn twists about separating curves

Abstract

Let $C$ be a curve in a closed orientable surface $F$ of genus $g\ge 2$ 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 $n\le 4g^2+2g$. We also give some additional applications.

21 pages, 4 figures

References in corpus (1)

• Dehn twists have roots

Cited by in corpus (2)

• General primitivity in the mapping class group
• Liftable mapping class groups of regular cyclic covers