Elliptic actions on Teichmüller space

  • Matthew Gentry Durham

    University of Michigan, Ann Arbor, USA
Elliptic actions on Teichmüller space cover
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Let be an oriented surface of finite type, its mapping class group, and its Teichmüller space with the Teichmüller metric. Let be a finite subgroup and consider the subset of fixed by , Fix. For any , we prove that the set of points whose -orbits have diameter bounded by , Fix, lives in a bounded neighborhood of Fix. As an application, we show that the orbit of any point has a fixed coarse barycenter, a property of nonpositive curvature. By contrast, we show that Fix 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 .

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

DOI 10.4171/GGD/494