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# Asian Journal of Mathematics

## Volume 20 (2016)

### Number 1

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

Pages: 157 – 182

DOI: http://dx.doi.org/10.4310/AJM.2016.v20.n1.a7

#### Authors

#### Abstract

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.

#### Keywords

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

Published 28 January 2016