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

Abstract

An action of a finite group $G$ is a pair $(S,\hat{G})$, where $S$ is a compact Riemann surface of genus $g⩾2$ and $\hat{G}⩽\mathrm{Aut}(S)$ is isomorphic to $G$. To each action $(S,\hat{G})$ there is associated a signature $(\gamma ;k_1,\dots ,k_r)$ 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 $\phi :S_1\to S_2$ such that $\phi G_1{\phi }^{-1}=G_2$. 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 $(0;k,\stackrel{n+1}{\dots },k)$.