Geodesic Length Functions and Teichmüller Spaces

Abstract

Given a compact orientable surface with finitely many punctures $\Sigma$, let $\Cal S(Σ)$ 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 $\Cal S(Σ)$ to $\mathbf{R}$ to be the geodesic length function of a hyperbolic metric with geodesic boundary and cusp ends on $\Sigma$. As a conse quence, the Teichmüller space of hyperbolic metrics with geodesic boundary and cusp ends on $\Sigma$ is reconstructed from an intrinsic $(\mathbf{Q}{P}^1,PSL(2,\mathbf{Z}))$ structure on $\Cal S(Σ)$.

32 pages, 13 figures