Geodesic Length Functions and Teichm\"uller Spaces

TL;DR

This paper characterizes the relations needed for geodesic length functions on hyperbolic surfaces with boundary and cusps, and reconstructs Teichmüller space from an intrinsic projective structure on simple closed curves.

Contribution

It provides a complete set of relations for geodesic length functions and reconstructs Teichmüller space from an intrinsic structure on simple closed curves.

Findings

Complete relations for geodesic length functions established.

Teichmüller space reconstructed from an intrinsic projective structure.

Provides a new perspective on hyperbolic metrics with boundary and cusps.

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 $\Cal S(\Sigma)$.