2022/07/28 by Pankaj Kapari, Kashyap Rajeevsarathy, Kapari, Pankaj +3
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Primary 57K20 #Secondary 57M60
paper · pdf · doi:10.48550/arxiv.2207.13910
openalex publication_date 2022/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 the existence of infinite metacyclic subgroups of Mod(Sg). 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 ℤ\rtimes ℤm, ℤn \rtimes ℤ, and ℤ \rtimes 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.