On the congruence subgroup property for GGS-groups

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 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.

Multi-GGS Groups have the Congruence Subgroup Property

“…Each group in the family is defined like a GGS group, except that there are more directed generators. In [3] we proved that a GGS group has the congruence subgroup property if and only if it is not defined by the constant vector. In this note, we show that the result is true for the whole family of multi-GGS groups.…”

“…For instance, the Gupta-Sidki group [5] is well known as a particularly uncomplicated answer to the general Burnside problem. It acts on the ternary rooted tree and is two-generated by a rooted automorphism and a directed one (see [3] for terminology and notation used here). A similar example, acting on the 4-regular rooted tree, was introduced by Grigorchuk in [4] and is now sometimes known as the 'second Grigorchuk group'.…”

“…As remarked previously, it suffices to show, by [3,Proposition 2.4] and Lemma 2, that G contains some level stabilizer. Lemma 5 and Proposition 6 yield that…”

The profinite completion of multi-EGS groups

“…The results of parts (B.1) and (C) are rather unexpected, when compared to part (A.1). The proofs of (A.1), (B.1) and (C) use a similar strategy as was done for the GGS-groups in [4,2,6] and for the EGS-groups in [13], though there are instances of new methods and ideas. The proof of (A.2) generalises the techniques used in [13], where the corresponding result was given for torsion EGS-groups.…”

Groups acting on rooted trees and their representations on the boundary

“…These groups attracted considerable interest recently and the reader is refered to the monograph [5, Sec. 2.3] and the articles [17,18,32] for more information. Here we consider (generalised) GGS-groups which act on a p k -regular rooted tree.…”