Comparing the real genus and the symmetric crosscap number of a group

Abstract

Given a finite group \(G\), there exist Klein surfaces, bordered \(X\) and unbordered non-orientable \(S\), such that \(G\) acts as an automorphism group of \(X\) and of \(S\). The minimum algebraic genus \(\rho(G)\) of the surfaces \(X\) is called the real genus of \(G\), and the minimal topological genus \(\tilde{\sigma}(G)\) of the surfaces \(S\) is the symmetric crosscap number of \(G\). In this work we study the relation between the real genus and the symmetric crosscap number of a group \(G\) and how both parameters can be compared. For instance, we see that there exist groups \(G\) such that the difference \(\tilde {\sigma} (G) - \rho (G) = t\) for all even negative numbers \(t\). In order to get it, we correct some inaccuracies in previous works, on these parameters for the groups \(C_m \times D_n\) and \(D_m \times D_n\). On the other hand, for some important families of groups, we prove that \(\tilde {\sigma} (G) = \rho(G) + 1\). We use it to eliminate possible gaps in the symmetric crosscap spectrum, enforcing the conjecture that \(3\) is in fact the unique gap.