Generating the liftable mapping class groups of cyclic covers of spheres

TL;DR

This paper provides finite generating sets and algorithms for presentations of liftable mapping class groups related to cyclic covers of spheres, with applications to normalizers, centralizers, and specific classes of covers.

Contribution

It introduces new generating sets and algorithms for presentations of liftable mapping class groups for cyclic covers, extending previous results and providing detailed structural insights.

Findings

Derived finite generating sets for liftable mapping class groups.

Developed algorithms for presentations, normalizers, and centralizers.

Recovered known generating sets for hyperelliptic and superelliptic covers.

Abstract

For $g\geq 2$, let $\mathrm{Mod}(S_g)$ be the mapping class group of closed orientable surface $S_g$ of genus $g$. In this paper, we derive a finite generating set for the liftable mapping class groups corresponding to finite-sheeted regular branched cyclic covers of spheres. As an application, we provide an algorithm to derive presentations of these liftable mapping class groups, and the normalizers and centralizers of periodic mapping classes corresponding to these covers. Furthermore, we determine the isomorphism classes of the normalizers of irreducible periodic mapping classes in $\mathrm{Mod}(S_g)$. Moreover, we derive presentations for the liftable mapping class groups corresponding to covers induced by certain reducible periodic mapping classes. Consequently, we derive a presentation for the centralizer and normalizer of a reducible periodic mapping class in $\mathrm{Mod}(S_g)$ of the highest order $2g+2$. As final applications of our results, we recover the generating sets of the liftable mapping class groups of the hyperelliptic cover obtained by Birman-Hilden and the balanced superelliptic cover obtained by Ghaswala-Winarski.