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

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

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26 Scopus citations

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.

Original languageEnglish
Pages (from-to)621-659
Number of pages39
JournalAnnales Academiae Scientiarum Fennicae Mathematica
Volume36
Issue number1
DOIs
StatePublished - 2011

Keywords

  • Fenchel-Nielsen metric
  • Fenchel-nielsen coordinates
  • Pair of pants decomposition
  • Quasiconformal metric
  • Surface of infinite type
  • Teichmüller metric
  • Teichmüller space

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