On the congruence subgroup property for GGS-groups

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 p , many examples of finitely generated, residually finite, non-torsion groups whose profinite completion is a pro- p 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.