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

TL;DR

This paper investigates the Cantor-Bendixson rank of the space of subgroups for certain self-replicating branch groups, specifically the Grigorchuk and Gupta-Sidki 3 groups, revealing it is always ω and characterizing subgroups by rank.

Contribution

It provides a detailed analysis of the Cantor-Bendixson rank for these groups and characterizes subgroups of each rank, advancing understanding of their subgroup structure.

Findings

The Cantor-Bendixson rank of Sub(G) is ω for the Grigorchuk and Gupta-Sidki 3 groups.

Subgroups of each finite rank n are characterized explicitly.

Descriptions of subgroups in the perfect kernel are provided.

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 $\mathrm{Sub}(G)$ is $\omega$. For each natural number $n$, we additionally characterize the subgroups of rank $n$ and give a description of subgroups in the perfect kernel.