Combinatorial density of the set of finitely maximal group actions of prime order on closed oriented surfaces

Suppose $X=X_g$ is a compact orientable surface of genus $g\ge 2$. A cyclic group $Z_p$ of prime order p of orientation-preserving homeomorphisms of X is called finitely maximal if there is no proper finite supergroup $G\le {\mathrm{Homeo}}^{+}\left(X\right)$ containing $Z_p$. Peterson et al. (2017) [8] showed that if $Z_p$ is finitely maximal then the number r of fixed points of $Z_p$ is maximal, or equivalently the genus h of $X/Z_p$ is minimal. Moreover, they exhibited an infinite sequence of genera within which any given action of $Z_p$ on X implies $Z_p$ is contained in some finite supergroup and demonstrate for genera outside of this sequence the existence of at least one finitely maximal $Z_p$-action. Here we show that the set of equivalence classes of finitely maximal cyclic actions of a fixed Teichmüller dimension d defined as $3(h-1)+r$ is combinatorially dense, in a suitable sense, in the set of classes of all actions of prime order of such dimension. This is a bit of a surprising result in the context of the results of Peterson, Russell and Wootton and it has some consequences concerning the singular locus of the moduli space of algebraic curves of a given genus, which we briefly describe at the end of the Introduction.