Mega-bimodules of topological polynomials: Sub-hyperbolicity and thurston obstructions

In 2006, Bartholdi and Nekrashevych solved a decade-old problem in holomorphic dynamics by creatively applying the theory of self-similar groups. Nekrashevych expanded this work in 2009 to define what we refer to as mega-bimodules which capture the topological data of Hurwitz classes of topological polynomials. He also showed that proving that these mega-bimodules are sub-hyperbolic will have two important implications: that all iterated monodromy groups of topological polynomials are contracting and that the Hubbard-Schliecher spider algorithm for complex polynomials generalizes to topological polynomials. We prove subhyperbolicity in the simplest non-trivial case and apply these mega-bimodules to holomorphic dynamics to prove a partial converse to the Berstein-Levy Theorem proved in 1985.