Geodesic length functions and teichmuller spaces

Abstract

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 consequence, the Teichmüller space of hyperbolic metrics with geodesic boundary and cusp ends on Σ is reconstructed explicitly from an intrinsic (QP1, PSL(2, Z)) structure on S(Σ). © 1998 Journal of Differential Geometry. © 1998 Applied Probability Trust.