Topological classification of Z_{p}^{m} actions on surfaces

Abstract

Let $\widetilde{S}$ be a closed (compact without boundary) oriented surface with genus $g$, and $G$ be a group isomorphic to $% \mathbf{Z}_{p}^{m}$, where $p$ is a prime integer. An action of $G$ on $S$ is a pair $(\widetilde{S},f)$, where $f$ is a representation of $G$ in the group of orientation preserving autohomeomorphisms of $\widetilde{S}$. Two actions $(\widetilde{S},f)$ and $(\widetilde{S^{\prime}},f^{\prime})$ are called strongly (resp. weakly) equivalent if there is a homeomorphism$,$ $% \widetilde{\psi}:\widetilde{S}\to \widetilde{S}^{\prime},$ sending the orientation of $\widetilde{S}$ to the orientation of $\widetilde{S}% ^{\prime},$ such that $f^{\prime}(h)=\widetilde{\psi}\circ f(h)\circ \widetilde{\psi}^{-1},$ (resp. there is an automorphism $\alpha \in Aut(G)$ such that $f^{\prime}\circ \alpha (h)=\widetilde{\psi}\circ f(h)\circ \widetilde{\psi}^{-1}$) for all $h\in G.$ We give the full description of strong and weak equivalence classes. The main idea of our work is the fact that a fixed point free action of $\mathbf{Z}_{p}^{m}$ on a surface provides a bilinear antisymmetric form on $\mathbf{Z}_{p}^{m}.$ For instance, we prove that the weakly equivalence classes of actions of $G$ on surfaces with orbit space of genus $g$ are in one to one correspondence with the set of pairs which consist in a positive integer number $k$, $k\leq m-n,$ $k=(m-n)% \func{mod}2,$ $g\geq {1/2}(m-n+k),$ and an orbit of the action of $% Aut(G)$ on the set of unordered $r$-tuples $[C_{1},...,C_{r}]$ of non-trivial elements generating a subgroup isomorphic to $\mathbf{Z}_{p}^{n}$ and such that $\sum_{1}^{r}C_{i}=0$. We use this result in describing the moduli space of complex algebraic curves admitting a group of automorphisms isomorphic to $\mathbf{Z}_{p}^{m}.$