Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 36, 2011, 621-659

ON FENCHEL-NIELSEN COORDINATES ON TEICHMÜLLER SPACES OF SURFACES OF INFINITE TYPE

Daniele Alessandrini, Lixin Liu, Athanase Papadopoulos, Weixu Su and Zongliang Sun

Max-Planck-Institut für Mathematik
Vivatsgasse 7, 53111 Bonn, Germany; daniele.alessandrini 'at' gmail.com

Sun Yat-sen (Zhongshan) University, Department of Mathematics
510275, Guangzhou, P.R. China; mcsllx 'at' mail.sysu.edu.cn

Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany, and
Université de Strasbourg and CNRS, Institut de Recherche Mathématique Avancée
7 rue René Descartes, 67084 Strasbourg Cedex, France; athanase.papadopoulos 'at' math.unistra.fr

Sun Yat-sen (Zhongshan) University, Department of Mathematics
510275, Guangzhou, P.R. China; su023411040 'at' 163.com

Shenzhen University, College of Mathematics and Computational Science
Shenzhen 518060, P.R. China; moonshoter 'at' 163.com

Abstract. We introduce Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type. The definition is relative to a given pair of pants decomposition of the surface. We start by establishing conditions under which any pair of pants decomposition on a hyperbolic surface of infinite type can be turned into a geometric decomposition, that is, a decomposition into hyperbolic pairs of pants. This is expressed in terms of a condition we introduce and which we call Nielsen-convexity. This condition is related to Nielsen cores of Fuchsian groups. We use this to define the Fenchel-Nielsen Teichmüller space relative to a geometric pair of pants decomposition. We study a metric, called the Fenchel-Nielsen metric, on such a Teichmüller space, and we compare it to the (quasiconformal) Teichmüller metric. We study conditions under which there is an equality between the Fenchel-Nielsen Teichmüller space and the familiar Teichmüller space defined using quasiconformal mappings, and we study topological and metric properties of the identity map between these two spaces when this map exists.

2010 Mathematics Subject Classification: Primary 32G15, 30F30, 30F60.

Key words: Surface of infinite type, pair of pants decomposition, Teichmüller space, Teichmüller metric, quasiconformal metric, Fenchel-Nielsen coordinates, Fenchel-Nielsen metric.

Reference to this article: D. Alessandrini, L. Liu, A. Papadopoulos, W. Su and Z. Sun: On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type. Ann. Acad. Sci. Fenn. Math. 36 (2011), 621-659.

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doi:10.5186/aasfm.2011.3637