TY - JOUR
T1 - On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type
AU - Alessandrini, Daniele
AU - Liu, Lixin
AU - Papadopoulos, Athanase
AU - Su, Weixu
AU - Sun, Zongliang
PY - 2011
Y1 - 2011
N2 - 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.
AB - 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.
KW - Fenchel-Nielsen metric
KW - Fenchel-nielsen coordinates
KW - Pair of pants decomposition
KW - Quasiconformal metric
KW - Surface of infinite type
KW - Teichmüller metric
KW - Teichmüller space
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U2 - 10.5186/aasfm.2011.3637
DO - 10.5186/aasfm.2011.3637
M3 - Article
AN - SCOPUS:84862684734
SN - 1239-629X
VL - 36
SP - 621
EP - 659
JO - Annales Academiae Scientiarum Fennicae Mathematica
JF - Annales Academiae Scientiarum Fennicae Mathematica
IS - 1
ER -