Elliptic actions on Teichmuller space

TL;DR

This paper studies the action of finite subgroups of the mapping class group on Teichmüller space, showing that points with bounded orbit diameter are close to fixed points, and explores properties of these fixed sets.

Contribution

It proves that points with bounded orbits under finite subgroup actions are near fixed points and introduces the concept of fixed coarse barycenters in Teichmüller space.

Findings

Points with bounded $H$-orbits are in a bounded neighborhood of $ ext{Fix}(H)$.

Every point's orbit under a finite order mapping class has a fixed coarse barycenter.

The set $ ext{Fix}_R^T(H)$ may not be quasiconvex, with explicit counterexamples.

Abstract

Let $S$ be an oriented surface of finite type, $\mathcal{MCG}(S)$ its mapping class group, and $\mathcal{T}(S)$ its Teichm\"uller space with the Teichm\"uller metric. Let $H \leq \mathcal{MCG}(S)$ be a finite subgroup and consider the subset of $\mathcal{T}(S)$ fixed by $H$, $\mathrm{Fix}(H) \subset \mathcal{T}(S)$. For any $R>0$, we prove that the set of points whose $H$-orbits have diameter bounded by $R$, $\mathrm{Fix}_R^T(H)$, lives in a bounded neighborhood of $\mathrm{Fix}(H)$. As an application, we show that the orbit of any point $X \in \mathcal{T}(S)$ under the action of a finite order mapping class has a fixed coarse barycenter. By contrast, we show that $\mathrm{Fix}^T_R(H)$ need not be quasiconvex with an explicit family of examples.