Parametrization of Teichmüller spaces by geodesic length functions

Abstract

The Teichmüller space T(Σ) of a compact C ^∞-surface Σ can be parametrized by geodesic length functions. More precisely, we can find a set {α1... ,α _n} of closed curves α j on Σ such that the isotopy class of a hyperbolic metric d on Σ (i.e. the point [d] ∊ T(Σ)) is determined by the lengths of geodesic curves homotopic to the curves α j on (Σ, d). However, since the fundamental group of Σ is not freely generated there is a quite complicated relation among these geodesic length function.