Abelianization of Symmetric Mapping Class Groups

2026/07/27 by Xiyan Zhong
Mathematics · #math.GT #math.AT

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Abstract

Let \widetildeS→ S be an unbranched regular p-fold cyclic cover of a closed orientable surface S of genus g. Two natural groups are associated with this cover. The first is the centralizer in Mod(\widetildeS) of a chosen generator σ of the deck transformation group, denoted by Mod(\widetildeS,σ). The second is the finite-index subgroup of Mod(S) consisting of mapping classes that fix the nonzero class [β]∈ H₁(S;ℤ/pℤ) corresponding to the cover, denoted by Mod(S,[β]). For p=2, the abelianizations of these groups were computed by Sato. We compute their abelianizations for every odd prime p and show that they exhibit a splitting phenomenon different from the case p=2. In most cases, this difference is reflected in the image of the Prym representation; in the remaining cases, it is detected by the existence of a distinguished element in the Johnson kernel.

Citations

• The abelianization of the level L mapping class group
• The abelianization of a symmetric mapping class group
• Fifteen problems about the mapping class groups
• Torelli Groups and Geometry of Moduli Spaces of Curves

Related

• Liftable mapping class groups of regular abelian covers
• The abelianization of a symmetric mapping class group
• Liftable mapping class groups of regular cyclic covers
• Liftable mapping class groups of certain branched covers of torus
• Arithmetic representations of mapping class groups