Infinite metacyclic subgroups of the mapping class group

Abstract

For g ≥ 2, let Mod(Sg) be the mapping class group of the closed orientable surface Sg of genus g. In this paper, we give a complete characterization of the infinite metacyclic subgroups of Mod(Sg) up to conjugacy. In particular, we provide equivalent conditions under which a pseudo-Anosov mapping class generates a metacyclic subgroup of Mod(Sg) with another mapping class. As applications of our main results, we establish the existence of infinite metacyclic subgroups of Mod(Sg) isomorphic to Z ⋊ k Zm, Zn ⋊ k Z, and Z ⋊ k Z. Furthermore, we derive bounds on the order of a periodic generator of an infinite metacyclic subgroup of Mod(Sg). Finally, we derive normal generating sets for the centralizers of irreducible periodic mapping classes.