Dynamics and Obstructions for Self-Similar Groups Generating Free Groups
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918137912360960 |
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| author | D'Angeli, Daniele Rodaro, Emanuele |
| author_facet | D'Angeli, Daniele Rodaro, Emanuele |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11048 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamics and Obstructions for Self-Similar Groups Generating Free Groups D'Angeli, Daniele Rodaro, Emanuele Group Theory 20E08, 20F65, 20E05, 68Q70 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. |
| title | Dynamics and Obstructions for Self-Similar Groups Generating Free Groups |
| topic | Group Theory 20E08, 20F65, 20E05, 68Q70 |
| url | https://arxiv.org/abs/2504.11048 |