Purely non-free actions on Riemann surfaces

Abstract

A group of conformal automorphisms of a Riemann surface is said to act purely non-free, or to produce a gpnf-action, if each of its finite-order elements has fixed points. It is known that every finite group admits a gpnf-action on a suitable closed Riemann surface of genus at least two. If $G$ is an infinite countable group, and $X$ is an infinite-type orientable surface, without planar ends and whose space of ends is self-similar, then we observe that there is a Riemann surface structure on $X$ admitting a gpnf-action by $G$. We obtain upper bounds for those finite groups producing a gpnf-action on a closed Riemann surface of genus $g\ge 2$. If $G$ is a finite group, then its purely non-free genus ${\sigma }_{hyp}^{pnf}(G)$ is the minimum genus of at least two of a closed Riemann surface admitting a gpnf-action by $G$. This parameter is already known for cyclic, abelian non-cyclic groups, ${\mathcal{A}}_4$, ${\mathcal{A}}_5$ and ${\mathfrak{S}}_4$. For general $G$, we provide the value of ${\sigma }_{hyp}^{pnf}(G)$, correcting a previous statement on this value. In particular, we observe that every genus is the purely non-free genus of some group. As an application, we describe the minimal purely non-free actions for the family of metacyclic groups $G_s^0(n)=⟨x,y:x^n=y^2=1,yxy=x^s⟩,$ where $n\ge 3$ and $2\le s\le n-1$ is such that $s^2\equiv 1\mathrm{mod}(n)$.