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

Frank Herrlich und Gabriela Schmithuesen

Erscheint in Handbook of Teichmueller theory, ed. A. Papadopoulos, European Mathematical Society

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