The symmetric genus spectrum of finite groups

Abstract

The symmetric genus of the finite group G , denoted by σ ( G ), is the smallest non-negative integer g such that the group G acts faithfully on a closed orientable surface of genus g (not necessarily preserving orientation). This paper investigates the question of whether for every non-negative integer g , there exists some G with symmetric genus g . It is shown that that the spectrum (range of values) of σ includes every non-negative integer g =!= 8 or 14 mod 18, and moreover, if a gap occurs at some g == 8 or 14 modulo 18, then the prime-power factorization of g − 1 includes some factor p e == 5 mod 6. In fact, evidence suggests that this spectrum has no gaps at all.