The symmetric genus of the Higman-Sims group \(HS\) and bounds for Conway's groups \(Co_ 1,Co_ 2\) (Q1186672)

Given a finite group \(G\), the symmetric genus \(\sigma(G)\) of \(G\) is defined to be the smallest integer \(g\) such that \(G\) acts faithfully on a closed orientable surface of genus \(g\). In this paper it is proved that \(\sigma(G)\leq 1+{1\over 2}| G| (1-{1\over 2}-{1\over 3}-{1\over 11})\) for \(G\) isomorphic to \(HS\), \(Co_ 1\) or \(Co_ 2\). Furthermore, equality is shown to obtain in the case of \(HS\).