Generating the liftable mapping class groups of regular cyclic covers

TL;DR

This work analyzes the liftable mapping class groups $\mathrm{LMod}_{p_k}(S_g)$ associated with the standard regular $k$-sheeted cyclic covers of a hyperbolic surface. By employing the symplectic representation and Johnson-style kernel results, the authors prove that $\mathrm{LMod}_{p_k}(S_g)$ is self-normalizing in $\mathrm{Mod}(S_g)$ and maximal when $k$ is prime, and they provide explicit finite generating sets for all $g\ge3$, $k\ge2$, including the $k=2$ case with special attention to the hyperelliptic involution. The paper also develops a detailed generating framework via $\Psi(\mathrm{Mod}_{p_k}(S_g,e_1))$ and its kernel, and extends these constructions to symmetric mapping class groups $\mathrm{SMod}_{p_k}(S_{g_k})$, as well as to liftings to infinite-type surfaces through the infinite ladder surface $\mathcal L$ with cover $q_g$. As applications, the results yield descriptions of the centralizer in $\mathrm{Mod}(S_g)$ of the hyperelliptic involution, the centralizer of a rotation in the two-sheeted case, and a finite generating set for $\mathrm{UMod}(S_g)=\mathrm{LMod}_{q_g}(S_g)$, highlighting connections between liftability, level structures, and big mapping class groups.

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 $\mathrm{LMod}_{p_2}(S_g) \cap C_{\mathrm{Mod}(S_g)}(ι)$, where $C_{\mathrm{Mod}(S_g)}(ι)$ is the centralizer of the hyperelliptic involution $ι\in \mathrm{Mod}(S_g)$. 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 $g \geq 3$.

Figures (8)

• Figure 1: A free regular cover $p_4: S_{g_4} \to S_g$ induced by a $\mathbb Z_4$-action on $S_{g_{4}}.$
• Figure 2: Each curve in the figure along with its image under the hyperelliptic involution $\iota$ forms a bounding pair that determines a map of type $F_i$.
• Figure 3: The curves $d_i$ in $S_g$.
• Figure 4: The Johnson chain of curves.
• Figure 6: The homomorphism induced $q_g$.

Theorems & Definitions (45)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 2.1: ADDR21
• Theorem 2.2
• Lemma 2.3
• Theorem 2.4
• proof
• Theorem 2.5