Perfect kernel of generalized Baumslag-Solitar groups
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915006091624448 |
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| author | Bontemps, Sasha |
| author_facet | Bontemps, Sasha |
| contents | In this article, we study the space of subgroups of generalized Baumslag-Solitar groups (GBS groups), that is, groups acting cocompactly on an oriented tree without inversion and with infinite cyclic vertex and edge stabilizers. Our results generalize the study of Baumslag-Solitar groups in [CGLMS22]. Given a GBS group G defined by a graph of groups whose existence is given by Bass-Serre theory, we associate to any subgroup of G an integer, which is a generalization of the phenotype defined in [CGLMS22]. This quantity is invariant under conjugation and allows us to decompose the perfect kernel of G into pieces which are invariant under conjugation and on which G acts highly topologically transitively. To achieve this, we interpret graphs of subgroups of G as "blown up and shrunk" Schreier graphs of transitive actions of G. We also describe the topology of the pieces which appear in the decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03221 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Perfect kernel of generalized Baumslag-Solitar groups Bontemps, Sasha Group Theory Dynamical Systems 37B, 20E06, 20E08 In this article, we study the space of subgroups of generalized Baumslag-Solitar groups (GBS groups), that is, groups acting cocompactly on an oriented tree without inversion and with infinite cyclic vertex and edge stabilizers. Our results generalize the study of Baumslag-Solitar groups in [CGLMS22]. Given a GBS group G defined by a graph of groups whose existence is given by Bass-Serre theory, we associate to any subgroup of G an integer, which is a generalization of the phenotype defined in [CGLMS22]. This quantity is invariant under conjugation and allows us to decompose the perfect kernel of G into pieces which are invariant under conjugation and on which G acts highly topologically transitively. To achieve this, we interpret graphs of subgroups of G as "blown up and shrunk" Schreier graphs of transitive actions of G. We also describe the topology of the pieces which appear in the decomposition. |
| title | Perfect kernel of generalized Baumslag-Solitar groups |
| topic | Group Theory Dynamical Systems 37B, 20E06, 20E08 |
| url | https://arxiv.org/abs/2411.03221 |