Dynamics and Obstructions for Self-Similar Groups Generating Free Groups
Group Theory
2025-09-10 v2
Abstract
We present obstruction results for self-similar groups regarding the generation of free groups. As a main consequence of our main results, we solve an open problem posed by Grigorchuk by showing that in an automaton group where a co-accessible state acts as the identity, any self-similar subgroup acting transitively on the first level, or any subgroup acting level-transitively on the rooted tree, is either cyclic or non-free. This result partially extends Sidki's findings for automaton groups with polynomial activity. Additionally, we show that for a reversible automaton group to generate a free group, the dual must necessarily contain a bireversible connected component.
Cite
@article{arxiv.2504.11048,
title = {Dynamics and Obstructions for Self-Similar Groups Generating Free Groups},
author = {Daniele D'Angeli and Emanuele Rodaro},
journal= {arXiv preprint arXiv:2504.11048},
year = {2025}
}