On the biggest purely non-free conformal actions on compact Riemann surfaces and their asymptotic properties

TL;DR

This paper investigates the maximal order of purely non-free conformal actions of finite groups on compact Riemann surfaces, establishing bounds, asymptotic behavior, and conjectures about their properties.

Contribution

It provides sharp bounds for the maximal order of gpnf-actions on surfaces of even genus and explores their asymptotic distribution, extending classical results.

Findings

Bound $oxed{8g}$ for even genus g is sharp for infinitely many g.

The set of accumulation points for even genus actions is 8.

For odd genus, the maximal order is between 4g and 8g, with conjectured sharpness.

Abstract

A continuous action of a finite group $G$ on a closed orientable surface $X$ is said to be gpnf (Gilman purely non-free) if every element of $G$ has a fixed point on $X$. We prove that the biggest order {$\mu(g)$}, of a gpnf-action on a surface of even genus $g \geq 2$, is bounded below by $8g$ and that this bound is sharp for infinitely many even $g$ as well. This provides, for even genera, a gpnf-action analog of the celebrated Accola-Maclachlan bound $8g+8$ for arbitrary finite continuous actions. We also describe the asymptotic behavior of $\mu$. We define $\mathcal{M}$ as the set of values of the form $$\widetilde{\mu}(g)=\frac{\mu(g)}{g+1},$$ and its subsets $\mathcal{M}_+$ and $\mathcal{M}_-$ corresponding to even and odd genera $g$. We show that the set $\mathcal{M}_+^d$, of accumulation points of $\mathcal{M}_+$, consists of a single number $8$. If $g$ is odd, then we prove that $4g \leq \mu(g)<8g$. We conjecture that this lower bound is sharp for infinitely many odd $g$. Finally, we prove that this conjecture implies that $4$ is the only element of $\mathcal{M}_-^d$, leading to $\mathcal{M}^d=\{4,8\}.$