Iterated Monodromy Groups of Intermediate Growth

Abstract

We give two new examples of groups of intermediate growth, by a method that was first used by Bux and Pérez. Our examples are the groups generated by the automata with the kneading sequences 11(0) and 0(011). By results of Nekrashevych, both of these groups are iterated monodromy groups of complex post-critically finite quadratic polynomials. We include a complete, systematic description of the Bux-Pérez method. We also prove, as a sample application of the method, that the groups determined by the automata with kneading sequence 1(0^k) (k >= 2) have intermediate growth, although this result is implicit in a survey article by Bartholdi, Grigorchuk, and Sunik. The paper concludes with an example of a group with no admissible length function; i.e., the group in question admits no length function with the properties required by the Bux-Pérez method. Whether the group has intermediate growth appears to be an open question.