Recognizing topological polynomials by lifting trees

TL;DR

This paper introduces an algorithm to identify whether a topological polynomial is equivalent to a complex polynomial, producing either the Hubbard tree or an obstruction, with applications to longstanding conjectures and problems.

Contribution

It presents a novel, geometric group theory-based algorithm for classifying topological polynomials and resolves several open problems in the field.

Findings

Algorithm successfully distinguishes polynomial equivalence

Resolved Pilgrim's finite global attractor conjecture

Provided new solutions to Hubbard's twisted rabbit problem

Abstract

We give a simple algorithm that determines whether a given post-critically finite topological polynomial is Thurston equivalent to a polynomial. If it is, 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 post-critical points is arbitrarily large.