# Elliptic actions on Teichmüller space

### Matthew Gentry Durham

University of Michigan, Ann Arbor, USA

## Abstract

Let $S$ be an oriented surface of finite type, $MCG(S)$ its mapping class group, and $T(S)$ its Teichmüller space with the Teichmüller metric. Let $H<MCG(S)$ be a finite subgroup and consider the subset of $T(S)$ fixed by $H$, Fix$(H)⊂T(S)$. For any $R>0$, we prove that the set of points whose $H$-orbits have diameter bounded by $R$, Fix$_{R}(H)$, lives in a bounded neighborhood of Fix$(H)$. As an application, we show that the orbit of any point $X∈T(S)$ has a fixed coarse barycenter, a property of nonpositive curvature. By contrast, we show that Fix$_{R}(H)$ 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 $MCG(S)$.

## Cite this article

Matthew Gentry Durham, Elliptic actions on Teichmüller space. Groups Geom. Dyn. 13 (2019), no. 2, pp. 415–465

DOI 10.4171/GGD/494