${\mathbb Z}_{k}^{m}$-actions of signature $(0;k,\stackrel{n+1}{\ldots},k)$

Abstract

An action of a finite group $G$ is a pair $(S,\hat{G})$, where $S$ is a compact Riemann surface of genus $g \geqslant 2$ and $\hat{G} \leqslant {\rm Aut}(S)$ is isomorphic to $G$. To each action $(S,\hat{G})$ there is associated a signature $(\gamma;k_{1},\ldots,k_{r})$ that codifies the orbifold structure of $S/\hat{G}$. Two actions of $G$, say $(S_{1},G_{1})$ and $(S_{2},G_{2})$, are topologically equivalent if there is an orientation-preserving homeomorphism $\varphi:S_{1} \to S_{2}$ such that $\varphi G_{1} \varphi^{-1}=G_{2}$. Topologically equivalent actions necessarily must have the same signature. The problem of determining the number of different topological actions of $G$ for a given signature is in general a difficult task. In this article, we describe, up to topological equivalence, those actions when $G$ is an abelian group and quotient genus $\gamma=0$. We are particularly interested in the case $G={\mathbb Z}_{k}^{m}$ and the quotient signature of the action to be of the form $(0;k,\stackrel{n+1}{\ldots},k)$.