  Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic
formulation of Thurston's characterization of rational functions. The
techniques developed are applied to the analysis of the dynamics on the set of
free homotopy classes of simple closed curves induced by a rational function.
The resulting finiteness results yield new information on the global dynamics
of the pullback map on Teichm\"uller space used in the proof of the
characterization theorem.
