The strong symmetric genus spectrum of abelian groups (Q527291)

The \textit{strong symmetric genus} \(\sigma^0(G)\) of a finite group \(G\) is the smallest genus of the compact Riemann surfaces on which \(G\) has a faithful action as a group of orientation-preserving automorphisms. It is known that every integer \(g \geq 0\) occurs as the strong symmetric genus of some \(G\), see [\textit{C. L. May} and \textit{J. Zimmerman}, Bull. Lond. Math. Soc. 35, No. 4, 433--439 (2003; Zbl 1028.57016)]. In this paper the authors completely determine the set \({\mathcal S}\) of all integers \(g \geq 0\) that occur as \(\sigma^0(G)\) for some abelian group \(G\). Indeed they show that \(g \in {\mathcal S}\) if and only if (a) \(g \equiv 1\) mod 4, or \(g \equiv 55\) mod 81, or (b) \(g-1\) is divisible by \(p^4\) for some odd prime \(p\), or (c) \(g-1\) is divisible by \(a^2\) for some odd integer \(a\) such that \(a-1\) divides \(g\), or (d) \(g-1\) is divisible by \(a^2(a-1)b^2\) for some odd integers \(a, b > 1\) with \(a \equiv 3\) mod 4. This has two interesting corollaries, namely that \({\mathcal S}\) contains no integer \(g \geq 2\) for which \(g-1\) is square-free, and that the range of values of \(\sigma^0\) on abelian groups of rank 5 or more is a subset of the range of \(\sigma^0\) on abelian groups of rank 3 or 4. The authors also prove that the set \({\mathcal S}\) has positive asymptotic density (in the set of non-negative integers), of approximately \(0.3284\).