The congruence subgroup problem for branch groups

Abstract

We state and study the congruence subgroup problem for groups acting on rooted tree, and for branch groups in particular. The problem is reduced to the computation of the congruence kernel, which we split into two parts: the branch kernel and the rigid kernel. In the case of regular branch groups, we prove that the first one is Abelian while the second has finite exponent. We also establish some rigidity results concerning these kernels. We work out explicitly known and new examples of non-trivial congruence kernels, describing in each case the group action. The Hanoi tower group receives particular attention due to its surprisingly rich behaviour.Comment: 22 pages, no figur

On the congruence subgroup property for GGS-groups

“…It would be interesting to know whether the pro-p completion of G coincides with its congruence completion, as well as to describe the congruence kernel of G. Previous work on the congruence subgroup problem for groups acting on rooted trees was done by Bartholdi, Siegenthaler and Zalesskii [3], where they developed tools to determine the congruence kernel of branch groups. However, these tools are not available to us, as the GGS-group with constant defining vector is not a branch group (although it is weakly branch).…”

The congruence subgroup problem for a family of branch groups

“…For notational purposes, we focus here on groups acting on regular rooted trees. A fuller discussion in the more general case of rooted spherically homogeneous trees can be found in [BSZ12], [Gar16b], or [Ski16].…”

“…The combination of Theorems 3.12, 3.15, and 3.17 show that unlike when n = 3, when n ≥ 4 the congruence kernel for G n is the same as the branch kernel. The following is extracted from the proof of Theorem 2.7 in [BSZ12].…”

“…In [BSZ12], it is shown the G 3 /G ′′ 3 is an infinite group and so G 3 is not just infinite. For n ≥ 4, the proof of Theorem 3.19 shows Rist Gn (m)/Rist Gn (m) ′ is finite.…”

“…Despite the existence of infinite families of groups having either trivial branch and trivial rigid kernel or non-trivial branch kernel but trivial rigid kernel, only one group appearing in the literature has been shown to have non-trivial rigid kernel. It is the Hanoi towers group on three pegs ( [BSZ12], [Ski16]), which we refer to as G 3 . In this paper, we study a family of generalizations of the Hanoi towers group, {G n | n ≥ 3}, and show that unlike G 3 , G n has trivial rigid kernel when n ≥ 4.…”