Recognizing topological polynomials by lifting trees

Abstract

We give a simple algorithm that determines whether a given postcritically finite topological polynomial is Thurston equivalent to a polynomial. If it is, then the algorithm produces the Hubbard tree; otherwise, the algorithm produces the canonical obstruction. Our approach is rooted in geometric group theory, using iteration on a simplicial complex of trees, and building on work of Nekrashevych. As one application of our methods, we resolve the polynomial case of Pilgrim’s finite global attractor conjecture. We also give a new solution to Hubbard’s twisted rabbit problem, and we state and solve several generalizations of Hubbard’s problem where the number of postcritical points is arbitrarily large.