Decorated Teichmüller Theory of Bordered Surfaces

Details Decorated Teichmüller Theory of Bordered Surfaces Decorated Teichmüller Theory of Bordered Surfaces

2002-10-21

arxiv.org/abs/math/0210326v2

Mathematics

Geometric Topology

This revision corrects an error in the application of the main theorem. The correct statement of the application is that the a

Scientific paper

This paper extends the decorated Teichm\"uller theory developed before for punctured surfaces to the setting of ``bordered'' surfaces, i.e., surfaces with boundary, and there is non-trivial new structure discovered. The main new result identifies the arc complex of a bordered surface up to proper homotopy equivalence with a certain quotient of the moduli space, namely, the quotient by the natural action of the positive reals by homothety on the hyperbolic lengths of geodesic boundary components. One tool in the proof is a homeomorphism between two versions of a ``decorated'' moduli space for bordered surfaces. The explicit homeomorphism relies upon points equidistant to suitable triples of horocycles.

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