A new approach to the genus spectra of abelian p-groups

Given a finite group G, the genus spetrum \(\textrm{sp}(G)\) $\text{sp}(G)$ of G is the set of integers \(g\ge 0\) $g\ge 0$ such that G can act faithfully on an orientable closed surface of genus g by orientation-preserving homeomorphisms. The determination of \(\textrm{sp}(G)\) $\text{sp}(G)$ is a classical topic and has a long history, but progress is lacked. In this paper, when G is an abelian p-group with \(p>2\) $p>2$ , we propose a new approach to \(\textrm{sp}(G)\) $\text{sp}(G)$ , giving a structural description for \(\textrm{sp}(G)\) $\text{sp}(G)$ in terms of a function which can be computed in finitely many steps.