Dynamics and Obstructions for Self-Similar Groups Generating Free Groups

2025/04/15 by Daniele D’Angeli, Emanuele Rodaro, D'Angeli, Daniele +1
Mathematics · #20E05 #20E08 #20F65 #68Q70 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2504.11048

openalex publication_date 2025/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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Related

• The conjugacy problem in automaton groups is not solvable
• Self-similar groups acting essentially freely on the boundary of the binary rooted tree
• Preserving self-similarity in free products of semigroups
• A characterization of those automata that structurally generate finite groups
• Automata and Square Complexes