Geometric realizations of cyclic actions on surfaces - II

Let $ text{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $ggeq 2$. Given a finite subgroup $H leq text{Mod}(S_g)$, let $text{Fix}(H)$ denote the set of fixed points induced by the action of $H$ on the Teichm{u}ller space $text{Teich}(S_g)$. The Nielsen realization problem, which was answered in the affirmative by S. Kerckhoff, asks whether $text{Fix}(H) eq emptyset$, for any given $H$. In this paper, we give an explicit description of $text{Fix}(H)$, when $H$ is cyclic. As consequences of our main result, we provide alternative proofs for two well known results, namely a result of Harvey on $text{dim}(text{Fix}(H))$, and a result of Gilman that characterizes irreducible finite order actions. Finally, we derive a correlation between the orders of irreducible cyclic actions and the filling systems on surfaces.