Geometric realizations of cyclic actions on surfaces -- II

TL;DR

This paper studies fixed point sets of cyclic and other finite subgroups acting on Teichmüller space of surfaces, revealing geometric decompositions, bounds on systoles, and connections to classical results in surface topology.

Contribution

It provides a detailed geometric analysis of fixed point sets for cyclic subgroups, including decompositions, systole bounds, and applications to classical theorems.

Findings

Fixed point sets decompose into products of strips with bounded width.

Derived a computable upper bound for systoles on fixed point sets.

Showed fixed point sets are not symplectomorphic to Euclidean space.

Abstract

Let $\mathrm{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. 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 Teichm\"{u}ller space $\mathrm{Teich}(S_g)$. When $H$ is cyclic with $|H| \geq 3$, 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 $\geq 3$, we derive a computable optimal upper bound for the restriction $\mathrm{sys} : \mathrm{Fix}(H) \to \mathbb{R}^+$ 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.