Geodesic Length Functions and Teichm\"uller Spaces

Abstract

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 Teichm\"uller 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 $\Cal S(\Sigma)$.