On the congruence subgroup property for GGS-groups

TL;DR

This paper proves that GGS-groups with non-constant defining vectors satisfy the congruence subgroup property, providing new examples of residually finite groups with specific profinite completions, and contrasts this with the constant vector case.

Contribution

It establishes the congruence subgroup property for non-constant GGS-groups and characterizes the constant case, answering a question of Barnea.

Findings

Non-constant GGS-groups satisfy the congruence subgroup property.

Constant GGS-group has an infinite congruence kernel and is not a branch group.

Provides examples of residually finite, non-torsion groups with pro-p profinite completions.

Abstract

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 torsion-free 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.