Geodesic length functions and Teichmüller spaces

. Given a compact orientable surface \Σ, let S (\Σ) be the set of isotopy classes of essential simple closed curves in \Σ. We determine a complete set of relations for a function from S (\Σ) to R to be the geodesic length function of a hyperbolic metric with geodesic boundary on \Σ. As a consequence, the Teichmuller space of hyperbolic metrics with geodesic boundary on \Σ is reconstructed from an intrinsic combinatorial structure on S(\Σ). This also gives a complete description of the image of Thurston's embedding of the Teichmuller space. x1. Introduction Let \Σ = \Σ g;r be a compact oriented surface of genus g with r boundary components. The Teichmuller space of isotopy classes of hyperbolic metrics with geodesic boundary on \Σ is denoted by T (\Σ), and the set of isotopy classes of essential simple closed unoriented curves in \Σ is denoted by S = S (\Σ). For each m 2 T (\Σ) and ff 2 S (\Σ), let l m (ff) be the length of t...