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

Abstract

We study the boundary of Teichmüller disks in T g , 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 T g 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 T g . 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.

The asymptotic behavior of Jenkins-Strebel rays

“…Augmented Teichmüller spaces. We refer to [Abi77] for augmented Teichmüller spaces; see also [IT92] and [HS07]. Let R be a connected Hausdorff space which satisfies the following conditions:…”

“…It is known (cf. [HS07]) that the Jenkins-Strebel rays r, r have limits, say r(∞), r (∞), on the boundary of the augmented Teichmüller spaceT (X). Suppose that r, r are similar, that is, there exist mutually disjoint simple closed curves γ 1 , · · · , γ k on X such that the set of homotopy classes of core curves of the annuli corresponding to q, q are represented by f (γ 1 ), · · · , f(γ k ) on Y and f (γ 1 ), · · · , f (γ k ) on Y respectively.…”

A comb of origami curves in the moduli space M 3 with three dimensional closure

“…The reader can find a detailed overview and further hints to literature e.g. in [7]. In this section we give a short introduction without proofs in order to provide the reader with the general ideas and explain everything that one needs to understand the results in Sects.…”

A global coordinate of the Teichmüller space related to asymptotic Jenkins–Strebel rays

“…The ray r = r(s) converges to the point r(∞) as s → ∞ for the topology ofT (for example, see Proposition 4.3 in [HS07]). Now, we determine the subset of the boundary ∂T which consists of Riemann surfaces with nodes obtained by pinching the given admissible curve family Γ, it is necessary to state Theorem 1.1.…”