On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type

Daniele Alessandrini (1) , Lixin Liu (2) , Athanase Papadopoulos (1, 3) , Weixu Su (2) , Zongliang Sun (4)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Department of Mathematics
3 MPI - Max-Planck-Institut für Mathematik
4 Department of Mathematics

We introduce Fenchel-Nielsen coordinates on Teicmüller 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 associated to a geometric pair of pants decomposition. We study a metric on such a Teichmüller space, and we compare it to the quasiconformal Teichmüller space, equipped with the Teichmüller metric. We study conditions under which there is an equality between these Teichmüller spaces and we study topological and metric properties of the identity map when this map exists.

Mots clés: Fenchel-Nielsen metric; surface of infinite type; pair of pants decomposition; Teichmüller space; Teichmüller metric; quasiconformal metric; Fenchel-Nielsen coordinates; Fenchel-Nielsen metric.

Domaines: Topologie géométrique [math.GT]

Daniele Alessandrini, Lixin Liu, Athanase Papadopoulos, Weixu Su, Zongliang Sun. On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type. Annales Academiae Scientiarum Fennicae. Mathematica, 2011, 36, p. 621-659. ⟨10.5186/aasfm.2011.3637⟩. ⟨hal-00461240⟩