Liftable mapping class groups of regular cyclic covers

Let $text{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g geq 1$. For $k geq 2$, 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 $text{LMod}_{p_k}(S_g)$ associated with the cover $p_k$. In particular, we show that $text{LMod}_{p_k}(S_g)$ is the stabilizer subgroup of $text{Mod}(S_g)$ 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 $text{LMod}_{p_k}(S_g)$, which generalizes of a well known normal series of congruence subgroups in $text{SL}(2,mathbb{Z})$. Among other applications, we describe a procedure for obtaining a finite generating set for $text{LMod}_{p_k}(S_g)$ and examine the liftability of certain finite-order and pseudo-Anosov mapping classes.