Liftable mapping class groups of regular abelian covers

TL;DR

This paper develops an algorithm to compute finite generating sets for liftable mapping class groups of regular abelian covers of surfaces, with applications to specific cyclic and abelian covers of genus 2 surfaces.

Contribution

It introduces a new algorithm for generating liftable mapping class groups of regular abelian covers, extending understanding of their structure and normalizers in the mapping class group.

Findings

Finite generating sets for liftable mapping class groups of certain covers.

Explicit generators for covers with cyclic and abelian deck groups.

Application of the algorithm to specific genus 2 surface covers.

Abstract

Let $S_g$ be the closed oriented surface of genus $g \geq 0$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. For $g\geq 2$, we develop an algorithm to obtain a finite generating set for the liftable mapping class group $\mathrm{LMod}_p(S_g)$ of a regular abelian cover $p$ of $S_g$. A key ingredient of our method is a result that provides a generating set of a group $G$ acting on a connected graph $X$ such that the quotient graph $X/G$ is finite. As an application of our algorithm, when $k$ is prime, we provide a finite generating set for $\mathrm{LMod}_{p_k}(S_2)$ for cyclic cover $p_k:S_{k+1}\to S_2$. Using the Birman-Hilden theory, when $k=2,3$ and $g=2$, we also obtain a finite generating set for the normalizer of the Deck transformation group of $p_k$ in $\mathrm{Mod}(S_{k+1})$. We conclude the paper with an application of our algorithm that gives a finite generating set for $\mathrm{LMod}_p(S_2)$, where $p:S_5\to S_2$ is a cover with deck transformation group isomorphic to $\mathbb{Z}_2\oplus \mathbb{Z}_2$.