Infinite metacyclic subgroups of the mapping class group

Abstract

For $g\ge 2$, let $\mathrm{Mod}(S_g)$ 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 a pair of elements in $\mathrm{Mod}(S_g)$ to generate an infinite metacyclic subgroup. In particular, we provide necessary and sufficient conditions under which a pseudo-Anosov mapping class generates an infinite metacyclic subgroup of $\mathrm{Mod}(S_g)$ with a nontrivial periodic mapping class. As applications of our main results, we establish the existence of infinite metacyclic subgroups of $\mathrm{Mod}(S_g)$ isomorphic to $\mathbb{Z}⋊{\mathbb{Z}}_m$, ${\mathbb{Z}}_n⋊\mathbb{Z}$, and $\mathbb{Z}⋊\mathbb{Z}$. Furthermore, we derive bounds on the order of a nontrivial periodic generator of an infinite metacyclic subgroup of $\mathrm{Mod}(S_g)$ that are realized. Finally, we show that the centralizer of an irreducible periodic mapping class $F$ is either $⟨F⟩$ or $⟨F⟩\times ⟨i⟩$, where $i$ is a hyperelliptic involution.