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Archiv der Mathematik

, Volume 112, Issue 2, pp 123–137 | Cite as

Pro-\(\mathcal {C}\) congruence properties for groups of rooted tree automorphisms

  • Alejandra GarridoEmail author
  • Jone Uria-Albizuri
Article
  • 11 Downloads

Abstract

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.

Keywords

Groups acting on rooted trees Weakly branch groups Congruence subgroups Profinite completions 

Mathematics Subject Classification

20E08 20E18 

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Notes

Acknowledgements

We are grateful to B. Klopsch for suggesting the problem and valuable discussions and to G. A. Fernández Alcober for suggesting several improvements. D. Francoeur, B. Klopsch, and H. Sasse pointed out an inaccuracy in [12] that affected our calculations (but not the main result) for the Basilica group in a previous version.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Mathematical and Physical SciencesUniversity of Newcastle AustraliaCallaghanAustralia
  2. 2.Basque Center of Applied MathematicsBilbaoSpain

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