An algebraic formulation of Thurston's combinatorial equivalence

• Published in: Proceedings of the American Mathematical Society Vol. 131; no. 11; pp. 3527 - 3534
• Main Author: Pilgrim, Kevin M.
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-11-2003
• Subjects: Algebra; Automorphisms; Closed curves; Critical points; Dynamical systems; Equivalence relation; Exact sciences and technology; Group theory; Group theory and generalizations; Homeomorphism; Mathematical theorems; Mathematics; Rational functions; Sciences and techniques of general use; Postcritically finite; endomorphism of free group; Postcritical set; Automorphism; Thurston combinatorial equivalence; Equivalence; Finite group; Free group endomorphism; Homeomorphism; Torus; Group theory; Orientation preserving covering; Combinatorics; Free group
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-03-07035-7

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.