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We give a method to construct surfaces of section for the geodisic flow on a negatively curved closed surface, with first return map semiconjugate to a toral Anosov diffeomorphism. This construction is used to prove that the discrete Godbillon Vey invariant [GS] is not a topological invariant, even when restricted to A variation on that construction produces examples of transitive Anosow flows related to [BL].

On the discrete Godbillon-Vey invariant and Dehn surgery on geodesic flows