On the congruence subgroup property for GGS-groups

Fernández-Alcober, Gustavo; Garrido, Alejandra; Uria-Albizuri, Jone

doi:10.1090/proc/13499

On the congruence subgroup property for GGS-groups
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by Gustavo A. Fernández-Alcober, Alejandra Garrido and Jone Uria-Albizuri PDF

Proc. Amer. Math. Soc. 145 (2017), 3311-3322 Request permission

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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