On purely non-free finite actions of abelian groups on compact surfaces (Q2411674)

A group \(G\) of self-homeomorphisms of a topological space \(X\) is said to act purely non-freely if each of its elements has a fixed point. If \(X\) is a surface, such actions are called gpnf-actions. First of all, the authors prove that, for any gnpf group \(G\) of order \(N\), there exists a Riemann surface of genus lying between \(N^2/4-2N+1\) and \(N^2/2-N+1\). In particular, there are cases where the minimal genus of a surface for a gpnf action of \(G\) lies in this range. Furthermore, the authors propose to study the maximal cyclic subgroups and the generators of a finite group \(G\) in order to find the minimal genus of a surface admitting a gpnf-action of \(G\).