Geodesic length functions and Teichmüller spaces

Given a compact orientable surface $\Sigma$, let $\mathcal {S}(\Sigma )$ be the set of isotopy classes of essential simple closed curves in $\Sigma$. We determine a complete set of relations for a function from $\mathcal {S}(\Sigma )$ to $\mathbf {R}$ to be the geodesic length function of a hyperbolic metric with geodesic boundary on $\Sigma$. As a consequence, the Teichmüller space of hyperbolic metrics with geodesic boundary on $\Sigma$ is reconstructed from an intrinsic combinatorial structure on $\mathcal {S}(\Sigma )$. This also gives a complete description of the image of Thurston's embedding of the Teichmüller space.