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Feb 13, 2017 at 18:46 comment added Richard Weidmann I think there is also a fairly elementary proof in "Automorphisms of Surfaces After Nielsen and Thurston" by Casson and Bleiler. The arguments should give you everything you need. Maybe nothing that isn't covered by Andy's references.
Feb 13, 2017 at 4:36 comment added Andy Putman I know two proofs of this for orientable surfaces (one using hyperbolic geometry that is explained in Farb-Margalit's primer on mapping class groups, the other using basic differential topology and explained in Buser's Geometry and Spectra of Compact Riemann Surfaces). Both proofs also work for non-orientable surfaces without changes.
Feb 13, 2017 at 1:24 history edited YCor
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Feb 13, 2017 at 1:01 history asked user104820 CC BY-SA 3.0