Commuting conjugates of finite-order mapping classes
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Commuting conjugates of finite-order mapping classes Neeraj K. Dhanwani1 · Kashyap Rajeevsarathy1 Received: 16 July 2019 / Accepted: 6 March 2020 © Springer Nature B.V. 2020
Abstract 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 for two finite-order mapping classes to have commuting conjugates in Mod(Sg ). 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 cyclic cover of Sg . Furthermore, we show that any nontrivial 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(Sg ) as isometry groups. Keywords Surface · Mapping class · Finite-order maps · Abelian subgroups Mathematics Subject Classification (2000) 57M60 · 57M50 · 57M99
1 Introduction Let Sg denote closed orientable surface of genus g ≥ 0, and let Mod(Sg ) denote the mapping class group of Sg . As we are only concerned with the finite abelian (non-cyclic) subgroups of Mod(Sg ), we will assume throughout the paper that g ≥ 2. Given two finite-order mapping classes in Mod(Sg ), for g ≥ 2, a natural question that arises is whether there exist representatives of their respective conjugacy classes that commute in Mod(Sg ). (When two finite-order mapping classes satisfy this condition, we say that they weakly commute.) While finite abelian groups and their conjugacy classes in Mod(Sg ) have been widely studied [2,4,9], our pursuit can be motivated with the following example. Consider the six involutions in Mod(S8 ) shown in Fig. 1, where each involution is realized as a π-rotation about an axis (passing through the origin) under a suitable isometric embedding S8 → R3 .
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Kashyap Rajeevsarathy [email protected] Neeraj K. Dhanwani [email protected]
1
Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal, Madhya Pradesh 462 066, India
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Geometriae Dedicata
z π
x
π
π
y
x
π x
y=x
Fig. 1 Six conjugate involutions in Mod(S8 )
Though all of these involutions are conjugate in Mod(Sg ), note that each of the two pairs of involutions indicated in the first two subfigures clearly generate distinct subgroups of Mod(S8 ) isomorphic to Z2 ⊕ Z2 , while the pair of involutions appearing in the third figure can be shown to generate a subgroup isomorphic to D8 . As the main result of this paper (see Theorem 4.1), in Sect. 4, we derive necessary and sufficient conditions under which two finite-order mapping classes will have commuting conjugates in Mod(Sg ). We appeal to Thurston’s orbifold theory [17], and the classical theory [4,5,8] of group actions on surfaces for proving this result. A key ingredient in our proof is un
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