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Algorithmic classification of surface homeomorphisms

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If you have a question about this talk, please contact Alexis Marchand.

Up to homotopy, homeomorphisms of closed surfaces come in three guises: periodic, reducible, and pseudo-Anosov. Among these three categories, pseudo-Anosov homeomorphisms exhibit qualitatively different topological, dynamical, and geometric properties.

The aim of this talk is to present an algorithm to decide if a surface homeomorphism is pseudo-Anosov, with a good theoretical upper bound on the running time. In particular, the algorithm runs in polynomial time in the genus of the surface and in the amount of information required to represent the input homeomorphism.

The inner workings of the algorithm rely on the combinatorics of splitting sequences of train tracks, together with a criterion of Masur and Minsky to estimate distances in the curve graph.

This talk is part of the Junior Geometry Seminar series.

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