Geodesic Length Functions and Teichmüller Spaces

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Given a compact orientable surface with finitely many punctures Σ, let S(Σ) be the set of isotopy classes of essential unoriented 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 and cusp ends on Σ. As a conse quence, the Teichmüller space of hyperbolic metrics with geodesic boundary and cusp ends on Σ is reconstructed from an intrinsic (QP¹, PSL(2, Z)) structure on S(Σ).