Split metacyclic actions on surfaces

TL;DR

This paper characterizes when two torsion elements in the mapping class group of a surface generate finite split metacyclic subgroups, classifies certain finite subgroups, and explores geometric realizations of these actions.

Contribution

It provides necessary and sufficient conditions for generating split metacyclic subgroups, classifies finite dihedral and quaternionic subgroups, and analyzes conjugacy and geometric realizations in the mapping class group.

Findings

Characterization of when torsion elements generate split metacyclic subgroups.

Complete classification of certain finite subgroups in $ ext{Mod}(S_g)$.

Existence of infinite dihedral subgroups generated by specific elements.

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 split non-abelian metacyclic subgroup of $\mathrm{Mod}(S_g)$. As applications of the main result, we give a complete characterization of the finite dihedral and the generalized quaternionic subgroups of $\mathrm{Mod}(S_g)$ up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that any finite-order mapping class whose corresponding orbifold is a sphere, has a conjugate that lifts under certain finite-sheeted regular cyclic covers of $S_g$. Moreover, for $g \geq 5$, we show the existence of an infinite dihedral subgroup of $\mathrm{Mod}(S_g)$ that is generated by an involution and a root of a bounding pair map of degree $3$. Finally, we provide a complete classification of the weak conjugacy classes of the non-abelian finite split metacyclic subgroups of $\mathrm{Mod}(S_3)$ and $\mathrm{Mod}(S_5)$. We also describe nontrivial geometric realizations of some of these actions.