2023/10/10 by Rajesh Dey, Kashyap Rajeevsarathy, Dey, Rajesh +1 · 1 citation
Mathematics · #Algebraic Geometry and Number 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.2310.06550
openalex publication_date 2023/10/10 · 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 of genus g ≥ 2. In this article, we derive necessary and sufficient conditions under which two torsion elements in Mod(Sg) will have conjugates that generate a finite symmetric or an alternating subgroup of Mod(Sg). Furthermore, we characterize when an involution would lift under the branched cover induced by an alternating action on Sg. Moreover, up to conjugacy, we derive conditions under which a given periodic mapping class is contained in a symmetric or an alternating subgroup of Mod(Sg). In particular, we show that symmetric or alternating subgroups can not contain irreducible mapping classes and hyperelliptic involutions. Finally, we classify the symmetric and alternating actions on S10 and S11 up to a certain equivalence we call weak conjugacy.