Geodesic Length Functions and Teichm\"uller Spaces

Given a compact orientable surface with finitely many punctures $\mathrm{\Sigma }$, let $\text{Cal}S(\mathrm{\Sigma })$ be the set of isotopy classes of essential unoriented simple closed curves in $\mathrm{\Sigma }$. We determine a complete set of relations for a function from $\text{Cal}S(\mathrm{\Sigma })$ to $\mathbf{R}$ to be the geodesic length function of a hyperbolic metric with geodesic boundary and cusp ends on $\mathrm{\Sigma }$. As a conse quence, the Teichm"uller space of hyperbolic metrics with geodesic boundary and cusp ends on $\mathrm{\Sigma }$ is reconstructed from an intrinsic $(\bold QP^1, PSL(2, \bold Z))$structureon$\Cal S(\Sigma)$.