Abelianization of Symmetric Mapping Class Groups

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 with this cover. The first is the centralizer in $\mathrm{Mod}(\widetilde{S})$ of a chosen generator $\sigma$ of the deck transformation group, denoted by $\mathrm{Mod}(\widetilde{S},\sigma)$. The second is the finite-index subgroup 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, denoted by $\mathrm{Mod}(S,[\beta])$. 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.