Liftable mapping class groups of regular cyclic covers

TL;DR

This paper studies liftable mapping class groups of cyclic covers of surfaces, characterizing their structure, criteria for liftability, and providing generators, with implications for understanding surface symmetries and congruence subgroups.

Contribution

It introduces a symplectic criterion for liftability, describes the structure of liftable groups as stabilizers, and generalizes known series of congruence subgroups.

Findings

Liftable groups are stabilizers of vectors in homology.

A symplectic criterion determines liftability of mapping classes.

A finite generating set for liftable groups is provided.

Abstract

Let $\mathrm{Mod}(S_g)$ 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 $\mathrm{LMod}_{p_k}(S_g)$ 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.