2020/07/16 by Neeraj K. Dhanwani, Kashyap Rajeevsarathy, Dhanwani, Neeraj K. +3
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Primary 57K20 #Secondary 57M60
paper · pdf · doi:10.48550/arxiv.2007.08279
openalex publication_date 2020/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Mod(Sg) be the mapping class group of the closed orientable surface Sg of genus g≥ 2. In this paper, we derive necessary and sufficient conditions under which two torsion elements in Mod(Sg) will have conjugates that generate a finite split non-abelian metacyclic subgroup of Mod(Sg). As applications of the main result, we give a complete characterization of the finite dihedral and the generalized quaternionic subgroups of Mod(Sg) up to a certain equivalence that we will call weak conjugacy. Furthermore, we show that any finite-order mapping class whose corresponding orbifold is a sphere, has a conjugate that lifts under certain finite-sheeted regular cyclic covers of Sg. Moreover, for g ≥ 5, we show the existence of an infinite dihedral subgroup of Mod(Sg) that is generated by an involution and a root of a bounding pair map of degree 3. Finally, we provide a complete classification of the weak conjugacy classes of the non-abelian finite split metacyclic subgroups of Mod(S3) and Mod(S5). We also describe nontrivial geometric realizations of some of these actions.