On the Cantor–Bendixson rank of the Grigorchuk group and the Gupta–Sidki 3 group

Abstract

We study the Cantor-Bendixson rank of the space of subgroups for members of a general class of finitely generated self-replicating branch groups. In particular, we show for G either the Grigorchuk group or the Gupta-Sidki 3 group, the Cantor-Bendixson rank of Sub(G) is ω. For each natural number n, we additionally characterize the subgroups of rank n and give a description of subgroups in the perfect kernel.

Subgroup induction property for branch groups

“…As a last application of the subgroup induction property, we expand a result of Skipper and Wesolek. See [14] for more and details on the Cantor-Bendixson rank. Proposition 3.13.…”

“…In recent years, the subgroup induction property of a branch group G turned out to be a versatile tool. For example, it was used in [14] to compute the Cantor-Bendixon rank of G and G 3 , in [11] to describe the weakly maximal subgroups (subgroups that are maximal for the property of being of infinite index) of G and in [9] to give a characterization of finitely generated subgroups of G, which was used to show that (if G has the congruence subgroup property and some other minor technical hypothesis) G is subgroup separable.…”

Finitely generated subgroups of branch groups and subdirect products of just infinite groups

“…It is self-replicating (also called fractal) if for every vertex v the section ϕ v (G) is, under the natural identification of T v to T , equal to G. Following [SW20] we will say that G is strongly self-replicating ). This is equivalent to the fact that the profinite topology on G coincide with the restriction on G of the natural topology of Aut(T ).…”