Annales Academi� Scientiarum Fennic�
Mathematica
Volumen 36, 2011, 621-659

ON FENCHEL-NIELSEN COORDINATES ON TEICHM�LLER SPACES OF SURFACES OF INFINITE TYPE

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

Max-Planck-Institut f�r 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 f�r Mathematik, Vivatsgasse 7, 53111 Bonn, Germany, and
Universit� de Strasbourg and CNRS, Institut de Recherche Math�matique Avanc�e
7 rue Ren� 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 Teichm�ller 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 Teichm�ller space relative to a geometric pair of pants decomposition. We study a metric, called the Fenchel-Nielsen metric, on such a Teichm�ller space, and we compare it to the (quasiconformal) Teichm�ller metric. We study conditions under which there is an equality between the Fenchel-Nielsen Teichm�ller space and the familiar Teichm�ller 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, Teichm�ller space, Teichm�ller 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 Teichm�ller 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