Abelianization of Symmetric Mapping Class Groups

Abstract

Let $\widetilde{S}\to S$ 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 $\mathrm{Mod}(\widetilde{S},\sigma)$ of a chosen generator $\sigma$ of the deck transformation group in $\mathrm{Mod}(\widetilde{S})$. The second is the finite-index subgroup $\mathrm{Mod}(S,[\beta])$ of $\mathrm{Mod}(S)$ consisting of mapping classes that fix the nonzero class $[\beta]\in H_1(S;\mathbb Z/p\mathbb Z)$ 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.