Quasiconformal Embeddings of Y-Pieces
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Quasiconformal Embeddings of Y-Pieces Peter Buser · Eran Makover · Björn Mützel · Robert Silhol
Received: 29 October 2013 / Revised: 19 January 2014 / Accepted: 24 January 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract In this paper we construct quasiconformal embeddings from Y-pieces that contain a short boundary geodesic into degenerate ones. These results are used in a companion paper to study the Jacobian tori of Riemann surfaces that contain small simple closed geodesics. Keywords
Quasiconformal maps · Pair of pants decomposition · Teichmüller space
Mathematics Subject Classification (2010)
30F30 · 30F45 · 30F60
Communicated by Matti Vuorinen. While working on this article, the third author was supported by the Alexander von Humboldt foundation. P. Buser (B) Department of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne, Switzerland e-mail: [email protected] E. Makover Department of Mathematics, Central Connecticut State University, 1615 Stanley Street, New Britain, CT, 06050, USA e-mail: [email protected] B. Mützel · R. Silhol Department of Mathematics, Université Montpellier 2, Eugène Bataillon, 34095 Montpellier cedex 5, France e-mail: [email protected] R. Silhol e-mail: [email protected]
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1 Introduction To study the behavior of geometric quantities in degenerating families of compact hyperbolic Riemann surfaces various types of standardized mappings have been devised in the literature that allow one to compare them with the corresponding quantities on the limiting surface. Colbois-Courtois [7], for instance, use suitably stretching maps of Y-pieces (see a few lines below) to prove convergence of the so-called small eigenvalues of the Laplacian, while Ji [10] investigates the large eigenfunctions with the help of the infinite energy harmonic maps from [12]. More recently, for the purpose of constructing quasiconformal deformations of Fuchsian groups with particular limit sets, Bishop [3,4] makes use of quasiconformal mappings of Y-pieces with exponential behavior in the thin ends that found further applications to quasiconformal mapping class groups, the augmented Teichmüller space and Riemann surfaces of infinite type, [1,2,8,11]. A Y-piece (or pair of pants) is a hyperbolic Riemann surface Y of signature (0, 3) whose boundary consists of closed geodesics and punctures. If the latter occur Y is called degenerate. Y-pieces serve as building blocks: all hyperbolic Riemann surfaces of finite area and many others may be built from them (see e.g. [6, Chapter 3]). In the present paper we construct quasiconformal embeddings of Y-pieces into degenerate and nearly degenerate ones, together with rather sharp dilatation estimates (Theorem 2.1, Theorem 5.1). The mappings can be extended in several ways to the Riemann surfaces in which the Y-pieces occur and will be used in [5] to study Jacobians of Riemann surfaces that lie close to the boundary of moduli space. By resorting to embeddings rather than surjections we are in a posi
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