Handbook of Teichmüller Theory, Volume I

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pp: 293–349

DOI: 10.4171/029-1/7

On the boundary of Teichmüller disks in Teichmüller and in Schottky space

Frank Herrlich[1] and Gabriela Schmithüsen[2]

(1) Karlsruhe Institute of Technology, Germany
(2) Universität Karlsruhe, Germany

We study the boundary of Teichmüller disks in Tg, a partial compactification of Teichmüller space, and their image in Schottky space.
  We give a broad introduction to Teichmüller disks and explain the relation between Teichmüller curves and Veech groups.
  Furthermore, we describe Braungardt’s construction of Tg and compare it with the Abikoff augmented Teichmüller space. Following Masur, we give a description of Strebel rays that makes it easy to understand their end points on the boundary of Tg. This prepares the description of boundary points that a Teichmüller disk has, with a particular emphasis to the case that it leads to a Teichmüller curve.
  Further on we turn to Schottky space and describe two different approaches to obtain a partial compactification. We give an overview how the boundaries of Schottky space, Teichmüller space and moduli space match together and how the actions of the diverse groups on them are linked. Finally we consider the image of Teichmüller disks in Schottky space and show that one can choose the projection from Teichmüller space to Schottky space in such a manner that the image of the Teichmüller disk is a quotient by an infinite group.

Keywords: Schottky space, Teichmüller disc, Veech group, Jenkins–Strebel ray, Schottky covering