We show that a contracting recurrent self-similar group of polynomial activity growth is amenable.  The proof proceeds by first showing that the associated limit space has conformal dimension equal to 1. We then show this implies the orbital graphs of its standard action are recurrent. Finally, we deduce amenability by applying results of Juschenko, Nekrashevych, and de la Salle.  Examples of such groups include the iterated monodromy groups of critically finite  hyperbolic “crochet” rational functions; these are maps for which any two points in the Fatou set are joined by a curve that meets the Julia set in an at most countable set.

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