Decorated Teichm\"uller Theory of Bordered Surfaces

Abstract

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.