On self-similar groups of intermediate growth

Abstract

We study several examples of self-similar groups of subexponential growth: the (generalized) Grigorchuk groups and the kneading automata groups induced by the sequences 1(10), 11(0) and 0(011). According to Nekrashevych, the last three groups appear as iterated monodromy groups for some complex post-critically finite quadratic polynomials. In particular, they support Nekrashevych’s conjecture on the intermediate growth of iterated monodromy groups.

For each of the cases we implement the method of incompressible elements. We conclude that the set of incompressible elements shares a common trait for all examples: the automaton described by the alternating patterns of incompressible elements consists of disjoint circles.