[Paper Review] On the congruence subgroup property for GGS-groups

Gustavo A. Fernández‐Alcober, Alejandra Garrido+1 more|arXiv (Cornell University)|Apr 12, 2016

Finite Group Theory Research14 references26 citations

TL;DR

This paper establishes that all GGS-groups with non-constant defining vectors satisfy the congruence subgroup property, providing new examples of finitely generated, residually finite, non-torsion groups whose profinite completion is a pro-p group—answering a question of Barnea. In contrast, the GGS-group with a constant defining vector has an infinite congruence kernel and is not a branch group, and its derived subgroup can be torsion-free under certain conditions on the defining vector.

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.

Motivation & Objective

• To determine whether GGS-groups with non-constant defining vectors satisfy the congruence subgroup property.
• To analyze the congruence kernel and group-theoretic structure of the GGS-group with constant defining vector.
• To answer Barnea’s question about the existence of infinite, finitely generated, residually finite, non-torsion groups with pro-p profinite completion, and whether such groups can be torsion-free.
• To identify conditions on the defining vector under which the derived subgroup of a GGS-group is torsion-free.

Proposed method

• Applying a general criterion by Bartholdi and Grigorchuk for regular branch groups to establish the congruence subgroup property for non-constant GGS-groups.
• Using the isomorphism ψn: stG(n) → Aut T × pn to analyze the structure of stabilizers and level components.
• Defining a property TF for vectors in Fp^{p-1} that ensures torsion-freeness of the derived subgroup via component analysis in the wreath product structure.
• Analyzing the action of the generator a and conjugates bi = b^ai to derive conditions on the product of components in ψ(h) for h ∈ stG(1).
• Employing circulant matrices C associated to the defining vector to compute component indices (m1,…,mp) and verify property TF.
• Using the fact that the congruence subgroup property is hereditary for finite index subgroups to reduce the torsion-freeness problem to the derived subgroup.

Experimental results

Research questions

• RQ1 Do all GGS-groups with non-constant defining vectors satisfy the congruence subgroup property?
• RQ2 Does the GGS-group with constant defining vector have an infinite congruence kernel?
• RQ3 Is the GGS-group with constant defining vector a branch group?
• RQ4 Can there exist infinite, finitely generated, residually finite, torsion-free groups whose profinite completion is a pro-p group?
• RQ5 Under what conditions on the defining vector is the derived subgroup of a GGS-group torsion-free?

Key findings

• All GGS-groups with non-constant defining vectors satisfy the congruence subgroup property, as proven via a general criterion for regular branch groups.
• The GGS-group with constant defining vector (e = (1,…,1)) has an infinite congruence kernel, implying its profinite completion differs from its congruence completion.
• This same GGS-group is not a branch group, despite being weakly branch, resolving a structural question about its self-similarity.
• For the GGS-group with defining vector e = (1,…,1,λ) and λ ≠ 1,2, the derived subgroup G′ is torsion-free, answering Barnea’s second question affirmatively.
• The derived subgroup G′ is infinite, finitely generated, residually finite, and has pro-p profinite completion when λ ≠ 1,2.
• A vector e ∈ Fp^{p-1} satisfies property TF if, for every non-zero (i1,…,ip) ∈ Fp^p with sum zero, there exists j such that mjij ≠ 0, where (m1,…,mp) = (i1,…,ip)C; if e satisfies TF, then G′ is torsion-free.