Generating the liftable mapping class groups of regular cyclic covers

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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-normalising 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 centraliser 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 $g \geq 3$ .