Thurston theory for critically fixed branched covering maps

Abstract: In the 1980’s, William Thurston obtained his celebrated characterisation of rational mappings. This result laid the foundation of such a field as Thurston's theory of holomorphic maps, which has been actively developing in the last few decades. One of the most important problems in this area is the questions about characterisation, which is understanding when a topological map is equivalent (in a certain dynamical sense) to a holomorphic one, and classification, which is an enumeration of all possible topological models of holomorphic maps from a given class.

In my talk, I am going to focus on the characterisation and classification problems for the family of post-critically finite branched coverings, i.e., branched coverings of the two-dimensional sphere $S^2$ with all critical points being fixed. Maps of this family can be defined by combinatorial models based on planar embedded graphs, and it provides an elegant answer to the classification problem for this family. Further, I plan to explain how to understand whether a given critically fixed branched cover is equivalent to a critically fixed rational map of the Riemann sphere and provide an algorithm of combinatorial nature that allows us to answer this question. Finally, if time permits, I will briefly mention the connections between Thurston's theory, Teichmüller spaces and Mapping Class Groups of marked spheres.

This is a joint work with Mikhail Hlushchanka.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

( paper | slides )

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Geometry and topology online

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The talks start at 13:30. Talks are typically fifty minutes long, with ten minutes for questions.

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