Pro-$\mathcal{C}$ congruence properties for groups of rooted tree automorphisms
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866917757978673152 |
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| author | Garrido, Alejandra Uria-Albizuri, Jone |
| author_facet | Garrido, Alejandra Uria-Albizuri, Jone |
| contents | We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro-$\mathcal{C}$ completions of the group, where $\mathcal{C}$ is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the $\mathcal{C}$-congruence subgroup property ($\mathcal{C}$-CSP) if its pro-$\mathcal{C}$ completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the $\mathcal{C}$-CSP. In the case where $\mathcal{C}$ is also closed under extensions (for instance the class of all finite $p$-groups for some prime $p$), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the $p$-CSP. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1709_03899 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Pro-$\mathcal{C}$ congruence properties for groups of rooted tree automorphisms Garrido, Alejandra Uria-Albizuri, Jone Group Theory 20E08 We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro-$\mathcal{C}$ completions of the group, where $\mathcal{C}$ is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the $\mathcal{C}$-congruence subgroup property ($\mathcal{C}$-CSP) if its pro-$\mathcal{C}$ completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the $\mathcal{C}$-CSP. In the case where $\mathcal{C}$ is also closed under extensions (for instance the class of all finite $p$-groups for some prime $p$), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the $p$-CSP. |
| title | Pro-$\mathcal{C}$ congruence properties for groups of rooted tree automorphisms |
| topic | Group Theory 20E08 |
| url | https://arxiv.org/abs/1709.03899 |