GROUPS OF EVEN REAL GENUS

Abstract

Let G be a finite group. The real genusρ (G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we develop some constructions of groups of even real genus, first using the notion of a semidirect product. As a consequence, we are able to show that for each integer g in certain congruence classes, there is at least one group of genus g. Next we consider the direct product Zn × G, in which one factor is cyclic and the other is a group of odd order that is generated by two elements. By placing a restriction on the genus action of G, we find the real genus of the direct product, in case n is relatively prime to |G|. We give some applications of this result, in particular to O*-groups, the odd order groups of maximum possible order. Finally we apply our results to the problem of determining whether there is a group of real genus g for each value of g. We prove that the set of integers for which there is a group has lower density greater than 5/6.