Decorated Teichmüller Theory of Bordered Surfaces
Abstract
Description
This paper extends the decorated Teichmüller 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.
This revision corrects an error in the application of the main theorem. The correct statement of the application is that the arc complex of a bordered surface is proper homotopy equivalent to a natural quotient of moduli space by the positive reals, where this group acts by scaling the hyperbolic lengths of all the geodesic boundary components
This revision corrects an error in the application of the main theorem. The correct statement of the application is that the arc complex of a bordered surface is proper homotopy equivalent to a natural quotient of moduli space by the positive reals, where this group acts by scaling the hyperbolic lengths of all the geodesic boundary components