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On the congruence subgroup property for GGS-groups


Authors: Gustavo A. Fernández-Alcober, Alejandra Garrido and Jone Uria-Albizuri
Journal: Proc. Amer. Math. Soc. 145 (2017), 3311-3322
MSC (2010): Primary 20E08
DOI: https://doi.org/10.1090/proc/13499
Published electronically: January 31, 2017
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Abstract: We show that all GGS-groups with a non-constant defining vector satisfy the congruence subgroup property. This provides, for every odd prime $ p$, many examples of finitely generated, residually finite, non-torsion groups whose profinite completion is a pro-$ p$ group, and among them we find torsion-free groups. This answers a question of Barnea. On the other hand, we prove that the GGS-group with a constant defining vector has an infinite congruence kernel and is not a branch group.


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Gustavo A. Fernández-Alcober
Affiliation: Department of Mathematics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain
Email: gustavo.fernandez@ehu.eus

Alejandra Garrido
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225, Düsseldorf, Germany
Email: alejandra.garrido@uni-duesseldorf.de

Jone Uria-Albizuri
Affiliation: Department of Mathematics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain
Email: jone.uria@ehu.eus

DOI: https://doi.org/10.1090/proc/13499
Received by editor(s): April 12, 2016
Received by editor(s) in revised form: September 14, 2016
Published electronically: January 31, 2017
Additional Notes: The first and third authors acknowledge financial support from the Spanish Government, grants MTM2011-28229-C02 and MTM2014-53810-C2-2-P, and from the Basque Government, grants IT753-13 and IT974-16. The third author was also supported by the Basque Goverment predoctoral grant PRE-2014-1-347. This article was finished while the second author was a postdoctoral researcher at the Université de Genève, whose support and that of the Swiss National Science Foundation she gratefully acknowledges.
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2017 American Mathematical Society