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Dynamics and Obstructions for Self-Similar Groups Generating Free Groups (2504.11048v1)

Published 15 Apr 2025 in math.GR

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.

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