We show that all GGS-groups with 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 torsionfree groups. This answers a question of Barnea. On the other hand, we prove that the GGS-group with constant defining vector has an infinite congruence kernel and is not a branch group.

The class of multi-EGS groups is a generalisation of the well-known Grigorchuk–Gupta–Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally, our results show that branch multi-EGS groups are just infinite.

If G is a Grigorchuk-Gupta-Sidki group defined over a p-adic tree, where p is an odd prime, we study the existence of Beauville surfaces associated to the quotients of G by its level stabilizers stG(n). We prove that if G is periodic then the quotients G/ stG(n) are Beauville groups for every n ≥ 2 if p ≥ 5 and n ≥ 3 if p = 3. In this case, we further show that all but finitely many quotients of G are Beauville groups. On the other hand, if G is non-periodic, then none of the quotients G/ stG(n) are Beauville groups.

We generalize the result about the congruence subgroup property for GGS-groups in [AGU] to the family of multi-GGS-groups; that is, all multi-GGS-groups except the one defined by the constant vector have the congruence subgroup property. Even if the result remains, new ideas are needed in order to generalize the proof.

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