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

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 $\mathcal{M}^d=\{4,8\}.$