The symmetric genus spectrum of finite groups

Abstract

The symmetric genus of the finite group G, denoted by σ(G), is the smallest nonnegative 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.

The symmetric genus spectrum of abelian groups

“…A natural problem is to determine the positive integers that occur as the symmetric genus of a group (or a particular type of group), that is, to determine the symmetric genus spectrum for the particular type of group. Important results about the symmetric genus spectrum of all finite groups were obtained by Conder and Tucker [1]. They showed that the symmetric genus spectrum of finite groups contains well over 88 percent of all positive integers.…”

“…They showed that the symmetric genus spectrum of finite groups contains well over 88 percent of all positive integers. In particular, they showed that if g is any non-negative integer such that g is not congruent to 8 or 14 (mod 18), then g is in the spectrum [1,Theorem 1.2]. However, there are no known gaps in the spectrum, and evidence suggests that there are none.…”

Groups of symmetric crosscap number less than or equal to 17

“…The symmetric genus 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). For this parameter, the spectrum includes every non-negative integer g ≡ 8 or 14 (mod 18), and moreover, if a gap occurs at some g ≡ 8 or 14 (mod 18), then the prime-power factorization of g − 1 includes some factor p e ≡ 5 (mod 6), [11]. In the study of the spectrum of the symmetric crosscap number, the groups with symmetric crosscap number of the form 12k + 3 are very interesting.…”

The Symmetric Genus of 2-Groups

“…The three automorphisms of order 2 of the maximal cyclic group will be called inversion, the quasi-dihedral action and the quasi-abelian action; these actions are given in (3), (4) and (5), respectively. Inversion is also used to construct a dicyclic group, but the element of the group that gives rise to the inner automorphism which is inversion has order 4.…”

“…Indeed, whether or not there is a group of symmetric genus n for each value of the integer n remains a challenging open question; see the recent, important article [4]. Here, we restrict our attention to 2-groups.…”