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 $g\ge 2$, 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 $$\tilde{\mu }(g)=\frac{\mu (g)}{g+1},$$ 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 $4g\le \mu (g)<8g$. 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 ${\mathcal{M}}^d=\{4,8\}.$