Geometric realizations of cyclic actions on surfaces: II

Abstract

Let $$\mathrm {Mod}(S_g)$$ denote the mapping class group of the closed orientable surface $$S_g$$ of genus $$g\ge 2$$ . Given a finite subgroup H of $$\mathrm {Mod}(S_g)$$ , let $$\mathrm {Fix}(H)$$ denote the set of fixed points induced by the action of H on the Teichmüller space $$\mathrm {Teich}(S_g)$$ . When H is cyclic with $$|H| \ge 3$$ , we show that $$\mathrm {Fix}(H)$$ admits a decomposition as a product of two-dimensional strips at least one of which is of bounded width. For an arbitrary H with at least one generator of order $$\ge 3$$ , we derive a computable optimal upper bound for the restriction $$\mathrm {sys}: \mathrm {Fix}(H) \rightarrow \mathbb {R}^+$$ of the systole function. Furthermore, we show that in such a case, $$\mathrm {Fix}(H)$$ is not symplectomorphic to the Euclidean space of the same dimension. Finally, we apply our theory to recover three well known results, namely: (a) Harvey’s result giving the dimension of $$\mathrm {Fix}(H)$$ , (b) Gilman’s result that H is irreducible if and only if the corresponding orbifold is a sphere with three cone points, and (c) the Nielsen realization theorem for cyclic groups.