Asian Journal of Mathematics

Volume 20 (2016)

Number 1

Hyperbolic metrics, measured foliations and pants decompositions for non-orientable surfaces

Pages: 157 – 182



A. Papadopoulos (Institut de Recherche Mathématique Avancée, Université de Strasbourg, France; and the Erwin Schrödinger International Institute for Mathematical Physics, Vienna, Austria)

R. C. Penner (Center for the Topology and Quantization of Moduli Spaces and Dept. of Math., Aarhus Univ., Aarhus, Denmark; Depts. of Math. and Physics, Caltech, Pasadena, Calif., USA; and the Erwin Schrödinger Int’l Institute for Mathematical Physics, Vienna, Austria)


We provide analogues for non-orientable surfaces with or without boundary or punctures of several basic theorems in the setting of the Thurston theory of surfaces which were developed so far only in the case of orientable surfaces. Namely, we provide natural analogues for non-orientable surfaces of the Fenchel–Nielsen theorem on the parametrization of the Teichmüller space of the surface, the Dehn–Thurston theorem on the parametrization of measured foliations in the surface, and the Hatcher–Thurston theorem, which gives a complete minimal set of moves between pair of pants decompositions of the surface. For the former two theorems, one in effect drops the twisting number for any curve in a pants decomposition which is 1-sided, and for the latter, two further elementary moves on pants decompositions are added to the two classical moves.


non-orientable surfaces, pants decompositions, Teichmüller space, Thurston’s boundary, Fenchel–Nielsen theorem, Dehn–Thurston theorem, Hatcher–Thurston theorem

2010 Mathematics Subject Classification

30F60, 32G15, 57M50, 57N16

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