Alternating and symmetric actions on surfaces

TL;DR

This paper characterizes when torsion elements in the mapping class group generate symmetric or alternating subgroups, explores lifting properties of involutions, and classifies such actions on surfaces of genus 10 and 11.

Contribution

It provides necessary and sufficient conditions for generating symmetric or alternating subgroups from torsion elements and classifies these actions on specific surfaces.

Findings

Torsion elements generate symmetric or alternating subgroups under specific conditions.

Involutions can be characterized for lifting under branched covers.

Classification of symmetric and alternating actions on surfaces of genus 10 and 11.

Abstract

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 2$. In this article, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite symmetric or an alternating subgroup of $\mathrm{Mod}(S_g)$. Furthermore, we characterize when an involution would lift under the branched cover induced by an alternating action on $S_g$. Moreover, up to conjugacy, we derive conditions under which a given periodic mapping class is contained in a symmetric or an alternating subgroup of $\mathrm{Mod}(S_g)$. In particular, we show that symmetric or alternating subgroups can not contain irreducible mapping classes and hyperelliptic involutions. Finally, we classify the symmetric and alternating actions on $S_{10}$ and $S_{11}$ up to a certain equivalence we call weak conjugacy.