Commuting conjugates of finite-order mapping classes

2019/01/31 by Dhanwani, Neeraj K., Rajeevsarathy, Kashyap
#57M99 #FOS: Mathematics #Geometric Topology (math.GT) #Primary 57M60 #Secondary 57M50

paper · doi:10.48550/arxiv.1901.11314

Abstract

Let Mod(S_g) be the mapping class group of the closed orientable surface S_g of genus g≥ 2. In this paper, we derive necessary and sufficient conditions for two finite-order mapping classes to have commuting conjugates in Mod(S_g). As an application of this result, we show that any finite-order mapping class, whose corresponding orbifold is not a sphere, has a conjugate that lifts under any finite-sheeted cover of S_g. Furthermore, we show that any torsion element in the centralizer of an irreducible finite order mapping class is of order at most 2. We also obtain conditions for the primitivity of a finite-order mapping class. Finally, we describe a procedure for determining the explicit hyperbolic structures that realize two-generator finite abelian groups of Mod(S_g) as isometry groups.

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