Iterated Monodromy Groups of Intermediate Growth

TL;DR

This paper introduces two new examples of groups with intermediate growth using a method by Bux and Pe9rez, and explores their properties as iterated monodromy groups of certain quadratic polynomials.

Contribution

It provides a systematic description of the Bux-Pe9rez method and applies it to new groups, also analyzing their growth and properties as iterated monodromy groups.

Findings

Two new groups of intermediate growth identified

Automata with specific kneading sequences generate these groups

Some groups lack admissible length functions, leaving their growth status open

Abstract

We give two new examples of groups of intermediate growth, by a method that was first used by Bux and P\'erez. 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-P\'erez 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-P\'erez method. Whether the group has intermediate growth appears to be an open question.