Geodesic Length Functions and Teichmüller Spaces

Details Geodesic Length Functions and Teichmüller Spaces Geodesic Length Functions and Teichmüller Spaces

1998-01-07

arxiv.org/abs/math/9801024v1

J. Differential Geometry, Vol. 48, 1998, 275-317.

Mathematics

Geometric Topology

32 pages, 13 figures

Scientific paper

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 $\Cal S(\Sigma)$.

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