Abelianization of Symmetric Mapping Class Groups

@misc{zhong2026abelianization,
  title = {{Abelianization of Symmetric Mapping Class Groups}},
  author = {Xiyan Zhong},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2607.24271},
  eprint = {2607.24271},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.24271},
}

TY  - UNPB
TI  - Abelianization of Symmetric Mapping Class Groups
AU  - Xiyan Zhong
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2607.24271
UR  - https://arxiv.org/abs/2607.24271
AB  - Let S̃→ S be an unbranched regular p-fold cyclic cover of a closed orientable surface S of genus g. Two natural groups are associated to this cover. The first is the centralizer Mod(S̃,σ) of a chosen generator σ of the deck transformation group in Mod(S̃). The second is the finite-index subgroup Mod(S,[β]) of Mod(S) consisting of mapping classes that fix the nonzero class [β]∈ H_1(S;ℤ/pℤ) corresponding to the cover. For p=2, Sato computed the abelianizations of these groups. We compute their abelianizations for every odd prime p and show that they exhibit a splitting phenomenon different from the case p=2. We also construct explicit abelianization maps using Morita's crossed homomorphism and the Prym representation.
ER  -