Geometric Analysis and Geometric Topologymathscidoc:1912.43782
arXiv preprint math/9801024, 1998.1
Given a compact orientable surface with finitely many punctures \Sigma , let $\Cal S (\Sigma) $ be the set of isotopy classes of essential unoriented simple closed curves in \Sigma . We determine a complete set of relations for a function from $\Cal S (\Sigma) $ to $\bold R $ to be the geodesic length function of a hyperbolic metric with geodesic boundary and cusp ends on \Sigma . As a conse quence, the Teichmller space of hyperbolic metrics with geodesic boundary and cusp ends on \Sigma is reconstructed from an intrinsic $(\bold QP^ 1, PSL (2,\bold Z)) $ structure on $\Cal S (\Sigma) $.