Metacyclic actions on surfaces

TL;DR

This paper characterizes when two torsion elements in the mapping class group generate finite metacyclic subgroups, providing classifications, bounds, and conditions for liftability, reducibility, and conjugacy of such actions on surfaces.

Contribution

It offers a complete characterization of non-split metacyclic subgroups in the mapping class group, including bounds, classifications, and conditions for liftability and reducibility.

Findings

4g is the upper bound for non-split metacyclic actions on S_g

Complete classification of dicyclic subgroups up to weak conjugacy

Every periodic element in a non-split metacyclic subgroup is reducible

Abstract

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have conjugates that generate a finite metacyclic subgroup of $\mathrm{Mod}(S_g)$. This yields a complete solution to the problem of liftability of periodic mapping classes under finite cyclic covers. As applications of the main result, we show that $4g$ is a realizable upper bound on the order of a non-split metacyclic action on $S_g$ and this bound is realized by the action of a dicyclic group. Moreover, we give a complete characterization of the dicyclic subgroups of $\mathrm{Mod}(S_g)$ up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that every periodic mapping class in a non-split metacyclic subgroup of $\mathrm{Mod}(S_g)$ is reducible. We provide necessary and sufficient conditions under which a non-split metacyclic action on $S_g$ factors via a split metacyclic action. Finally, we provide a complete classification of the weak conjugacy classes of the finite non-split metacyclic subgroups of $\mathrm{Mod}(S_{10})$ and $\mathrm{Mod}(S_{11})$.