Elliptic actions on Teichmüller space

Abstract

Let $S$ be an oriented surface of finite type, $\mathcal{MC}\mathcal{G}(S)$ its mapping class group, and $\mathcal{T}(S)$ its Teichmüller space with the Teichmüller metric. Let $H<\mathcal{MC}\mathcal{G}(S)$ be a finite subgroup and consider the subset of $\mathcal{T}(S)$ fixed by $H$, Fix$(H)\subset \mathcal{T}(S)$. For any $R>0$, we prove that the set of points whose $H$-orbits have diameter bounded by $R$, Fix${}_R^T(H)$, lives in a bounded neighborhood of Fix$(H)$. As an application, we show that the orbit of any point $X\in \mathcal{T}(S)$ has a fixed coarse barycenter, a property of nonpositive curvature. By contrast, we show that Fix${}_R^T(H)$ need not be quasiconvex with an explicit family of examples. As an application of the barycenter theorem, we prove that there is an exponential-time algorithm to solve the conjugacy problem for finite order subgroups of $\mathcal{MC}\mathcal{G}(S)$.