Recognizing Topological Polynomials by Lifting Trees

Jim Belk (University of St Andrews)

Monday 3rd February 16:00-17:00 Maths 311B

Abstract

A topological polynomial is an orientation-preserving branched cover of the plane by itself with finitely many branch points. In a celebrated theorem, Thurston proved that any marked topological polynomial is either topologically equivalent to a complex polynomial or has a geometric obstruction known as a Thurston obstruction. In joint work with Justin Lanier, Dan Margalit, and Becca Winarski, we describe a simple geometric algorithm to determine whether a given branched cover is equivalent to a complex polynomial and if so, which one. Our methods are rooted in geometric group theory, and involve a non-expanding map on a simplicial complex whose vertices correspond to finite topological trees.

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