Abelian $p$-Groups of Symmetries of Surfaces

Abstract

An integer $g \geq 2$ is said to be a genus of a finite group $G$ if there is a compact Riemann surface of genus $g$ on which $G$ acts as a group of automorphisms. In this paper finite abelian $p$-groups of arbitrarily large rank, where $p$ is an odd prime, are investigated. For certain classes of abelian $p$-groups the minimum reduced stable genus $\sigma_0$ of $G$ is calculated and consequently the genus spectrum of $G$ is completely determined for certain "extremal" abelian $p$-groups. Moreover for the case of $Z_p^{r_1} \oplus Z_{p^2}^{r_2}$ we will see that the genus spectrum determines the isomorphism class of the group uniquely.