GGS-groups: Order of congruence quotients and Hausdorff dimension

Abstract

If G is a GGS-group defined over a p-adic tree, where p is an odd prime, we calculate the order of the congruence quotients Gn = G/ StabG(n) for every n. If G is defined by the vector e = (e1, . . . , ep−1) ∈ F p−1 p , the determination of the order of Gn is split into three cases, according as e is non-symmetric, non-constant symmetric, or constant. The formulas that we obtain only depend on p, n, and the rank of the circulant matrix whose first row is e. As a consequence of these formulas, we also obtain the Hausdorff dimension of the closures of all GGS-groups over the p-adic tree.

Maximal subgroups of multi-edge spinal groups

“…The next result mimics [4,Lemma 3.2], which applies to GGS-groups. We remark that there are no new exceptions, in addition to the GGS-group…”

“…We do not know a proof that the GGS-group G in (3.7) is not branch. From properties that were established in [4] we derive the following result. …”

Abstract commensurability and the Gupta–Sidki group

“…This property will be important for the proofs of Theorem 1 and Theorem 5 but it is also obviously useful for the study of finite quotients of G. In this direction, we point out that in [3] the authors give an explicit formula for the indices |Γ : St Γ (n)| not just for the Gupta-Sidki p-group, but a more general class of groups Γ which act on p-regular rooted trees (GGS groups). They also prove, using different methods, some of the properties stated in the previous lemmas for arbitrary GGS groups.…”

Engel elements in weakly branch groups

“…Thus in the remainder we assume that e is not constant. By Lemmas 3.2 and 3.4 of [8], we know that G is regular branch over K, where K = γ 3 (G) if e is symmetric and K = G otherwise. Since rst G (n) contains a copy of K × p n · · · × K for every n ∈ N, if we prove that K is not Engel then Theorem C applies to conclude that R(G) = 1.…”