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

Abstract

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.

Curve graphs on surfaces of infinite type

“…To be precise, we require that our surfaces have a certain type of bounded geometry, namely that they admit a (generalized) pants decomposition where supremum of the lengths of the individual curves in the decomposition is bounded. Deformation spaces of such surfaces have been studied by Alessandrini et al [1,2] and have been called upper-bounded surfaces.…”

Thurston's metric on Teichmüller space and the translation distances of mapping classes

“…Furthermore, we can glue all the new pairs of pants together (in the same topological pattern as before) with appropriate twists such that the resulting structure is f (X j L ). The gluing map is again a quasiconformal mapping, with dilatation controlled by (an upper bound of) the lengths and twists of the curves in the pants decomposition (see [3]). …”

On metrics defined by length spectra on Teichmüller spaces of surfaces with boundary

“…Otherwise it is said to be of infinite type. For more details on Teichmüller spaces of surfaces of infinite type (where the definition of Teichmüller space is not unique and more involved), we refer to [3].…”