Liftable mapping class groups of regular cyclic covers

Abstract

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g\ge 1$. For $k\ge 2$, we consider the standard $k$-sheeted regular cover $p_k:S_{k(g-1)+1}\to S_g$, and analyze the liftable mapping class group ${\mathrm{LMod}}_{p_k}(S_g)$ associated with the cover $p_k$. In particular, we show that ${\mathrm{LMod}}_{p_k}(S_g)$ is the stabilizer subgroup of $\mathrm{Mod}(S_g)$ with respect to a collection of vectors in $H_1(S_g,{\mathbb{Z}}_k)$, and also derive a symplectic criterion for the liftability of a given mapping class under $p_k$. As an application of this criterion, we obtain a normal series of ${\mathrm{LMod}}_{p_k}(S_g)$, which generalizes a well known normal series of congruence subgroups in $\mathrm{SL}(2,\mathbb{Z})$. Among other applications, we describe a procedure for obtaining a finite generating set for ${\mathrm{LMod}}_{p_k}(S_g)$ and examine the liftability of certain finite-order and pseudo-Anosov mapping classes.