An algebraic formulation of Thurston’s combinatorial equivalence

Abstract:

Let $f:S^2 \to S^2$ be an orientation-preserving branched covering for which the set $P_f$ of strict forward orbits of critical points is finite and let $G=\pi _1(S^2-f^{-1}P_f)$. To $f$ we associate an injective endomorphism $\varphi _f$ of the free group $G$, well-defined up to postcomposition with inner automorphisms. We show that two such maps $f,g$ are combinatorially equivalent (in the sense introduced by Thurston for the characterization of rational functions as dynamical systems) if and only if $\varphi _f, \varphi _g$ are conjugate by an element of $\operatorname {Out}(G)$ which is induced by an orientation-preserving homeomorphism.