Infinite metacyclic subgroups of the mapping class group

For g≥ 2 , let Mod(Sg) be the mapping class group of the closed orientable surface Sg of genus g . In this paper, we provide necessary and sufficient conditions for a pair of elements in Mod(Sg) 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 Mod(Sg) with a nontrivial periodic mapping class. As applications of our main results, we establish the existence of infinite metacyclic subgroups of Mod(Sg) isomorphic to Zₘ , Zₙ , and Z . Furthermore, we derive bounds on the order of a nontrivial periodic generator of an infinite metacyclic subgroup of Mod(Sg) that are realized. Finally, we show that the centralizer of an irreducible periodic mapping class F is either F or F× i , where i is a hyperelliptic involution.