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

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 $\mu \left(g\right)$, 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 μ. We define $M$ as the set of values of the function $\tilde{\mu }\left(g\right)=\mu \left(g\right)/(g+1)$ and its subsets $M_{+}$ and $M_{-}$ corresponding to even and odd genera g. We show that the set $M_{+}^d$, of accumulation points of $M_{+}$, consists of a single number 8. If g is odd, then we prove that $4g\le \mu \left(g\right)<8g$. 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 $M_{-}^d$, leading to $M^d=\{4,8\}$.