Communications in Analysis and Geometry

Volume 12 (2004)

Number 4

Decorated Teichmüller theory of bordered surfaces

Pages: 793 – 820



R. C. Penner


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. Beyond this, the main new result of this paper identifies an open dense subspace of 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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