AI 中文总结
研究关于闭可定向曲面无分支正则$p$重循环覆盖相关的两个自然群，计算奇素数$p$时它们的交换化，发现与$p = 2$时不同的分裂现象，差异体现在普里姆表示像或约翰逊核特殊元素中。
AI 中文摘要
设$\widetilde{S}\to S$是亏格为$g$的闭可定向曲面$S$的无分支正则$p$重循环覆盖。与此覆盖相关联有两个自然群。第一个是覆盖变换群的选定生成元$\sigma$在$\mathrm{Mod}(\widetilde{S})$中的中心化子，记为$\mathrm{Mod}(\widetilde{S},\sigma)$。第二个是$\mathrm{Mod}(S)$中由固定与覆盖对应的非零类$[\beta]\in H_1(S;\mathbb{Z}/p\mathbb{Z})$的映射类组成的有限指数子群，记为$\mathrm{Mod}(S,[\beta])$。对于$p = 2$，佐藤计算了这些群的交换化。我们计算了每个奇素数$p$时它们的交换化，并表明它们呈现出与$p = 2$的情况不同的分裂现象。在大多数情况下，这种差异反映在普里姆表示的像中；在其余情况下，通过约翰逊核中一个特殊元素的存在来检测。
英文摘要
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 $\mathrm{Mod}(\widetilde{S},σ)$ of a chosen generator $σ$ of the deck transformation group in $\mathrm{Mod}(\widetilde{S})$. The second is the finite-index subgroup $\mathrm{Mod}(S,[β])$ of $\mathrm{Mod}(S)$ consisting of mapping classes that fix the nonzero class $[β]\in H_1(S;\mathbb Z/p\mathbb Z)$ 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.
Comments21 pages, 1 figure. Replaced the technical computation of the Johnson homomorphism in the final step of the proof of Theorem 1.1 with an explicit construction of the abelianization map
