Liftable mapping class groups of regular abelian covers

Abstract

Let $S_g$ be the closed oriented surface of genus $g\ge 0$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. For $g\ge 2$, we develop an algorithm to obtain a finite generating set for the liftable mapping class group ${\mathrm{LMod}}_p(S_g)$ of a regular abelian cover $p$ of $S_g$. A key ingredient of our method is a result that provides a generating set of a group $G$ acting on a connected graph $X$ such that the quotient graph $X/G$ is finite. As an application of our algorithm, when $k$ is prime, we provide a finite generating set for ${\mathrm{LMod}}_{p_k}(S_2)$ for cyclic cover $p_k:S_{k+1}\to S_2$. Using the Birman-Hilden theory, when $k=2,3$ and $g=2$, we also obtain a finite generating set for the normalizer of the Deck transformation group of $p_k$ in $\mathrm{Mod}(S_{k+1})$. We conclude the paper with an application of our algorithm that gives a finite generating set for ${\mathrm{LMod}}_p(S_2)$, where $p:S_5\to S_2$ is a cover with deck transformation group isomorphic to ${\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$.

27 pages, 9 figures