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 ≥ 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 ≤ Homeo + ( X ) containing Z p . In [Maximal group actions on compact oriented surfaces, J. Algebra 472 (2017) 1–14], V. Peterson, J. Russell, and A. Wootton 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.