Infinite metacyclic subgroups of the mapping class group

TL;DR

This paper characterizes when infinite metacyclic subgroups exist in the mapping class group of surfaces, especially involving pseudo-Anosov and periodic elements, and provides bounds and structural descriptions of these subgroups.

Contribution

It offers necessary and sufficient conditions for the existence of infinite metacyclic subgroups in the mapping class group, including classifications involving pseudo-Anosov and periodic mapping classes.

Findings

Conditions for existence of infinite metacyclic subgroups

Classification of subgroups involving pseudo-Anosov and periodic elements

Bounds on orders of periodic generators

Abstract

For $g\geq 2$, let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we provide necessary and sufficient conditions for the existence of infinite metacyclic subgroups of $\text{Mod}(S_g)$. In particular, we provide necessary and sufficient conditions under which a pseudo-Anosov mapping class generates an infinite metacyclic subgroup of $\text{Mod}(S_g)$ with a nontrivial periodic mapping class. As applications of our main results, we establish the existence of infinite metacyclic subgroups of $\text{Mod}(S_g)$ isomorphic to $\mathbb{Z}\rtimes \mathbb{Z}_m, \mathbb{Z}_n \rtimes \mathbb{Z}$, and $\mathbb{Z} \rtimes \mathbb{Z}$. Furthermore, we derive bounds on the order of a nontrivial periodic generator of an infinite metacyclic subgroup of $\text{Mod}(S_g)$ that are realized. Finally, we show that the centralizer of an irreducible periodic mapping class $F$ is either $\langle F\rangle$ or $\langle F\rangle \times \langle i\rangle$, where $i$ is a hyperelliptic involution.