Generating the 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 $g \geq 1$, and let $\mathrm{LMod}_{p}(X)$ be the liftable mapping class group associated with a finite-sheeted branched cover $p:S \to X$, where $X$ is a hyperbolic surface. For $k \geq 2$, let $p_k: S_{k(g-1)+1} \to S_g$ be the standard $k$-sheeted regular cyclic cover. In this paper, we show that $\{\mathrm{LMod}_{p_k}(S_g)\}_{k \geq 2}$ forms an infinite family of self-normalizing subgroups in $\mathrm{Mod}(S_g)$, which are also maximal when $k$ is prime. Furthermore, we derive explicit finite generating sets for $\mathrm{LMod}_{p_k}(S_g)$ for $g \geq 3$ and $k \geq 2$, and $\mathrm{LMod}_{p_2}(S_2)$. For $g \geq 2$, as an application of our main result, we also derive a generating set for $\mathrm{LMod}_{p_2}(S_g) \cap C_{\mathrm{Mod}(S_g)}(\iota)$, where $C_{\mathrm{Mod}(S_g)}(\iota)$ is the centralizer of the hyperelliptic involution $\iota \in \mathrm{Mod}(S_g)$. Let $\mathcal{L}$ be the infinite ladder surface, and let $q_g : \mathcal{L} \to S_g$ be the standard infinite-sheeted cover induced by $\langle h^{g-1} \rangle$ where $h$ is the standard handle shift on $\mathcal{L}$. As a final application, we derive a finite generating set for $\mathrm{LMod}_{q_g}(S_g)$ for $g \geq 3$.