Quasi-abelian group as automorphism group of Riemann surfaces

Abstract

Conformal/anticonformal actions of the quasi-abelian group \(QA_{n}\) of order \(2^n\), for \(n\ge 4\), on closed Riemann surfaces, pseudo-real Riemann surfaces and closed Klein surfaces are considered. We obtain several consequences, such as the solution of the minimum genus problem for the \(QA_n\)-actions, and for each of these actions, we study the topological rigidity action problem. In the case of pseudo-real Riemann surfaces, attention was typically restricted to group actions that admit anticonformal elements. In this paper, we consider two cases: either \(QA_n\) has anticonformal elements or only contains conformal elements.