Finitely generated subgroups of branch groups and subdirect products of just infinite groups
Rostistlav Grigorchuk, Paul-Henry Leemann, Tatiana Nagnibeda
TL;DR
The paper addresses how to characterize finitely generated subgroups of branch groups, notably the Grigorchuk group and Gupta-Sidki p-groups, by introducing block subgroups and analyzing subdirect products of just infinite groups. It develops a general framework showing that finitely generated subgroups are controlled by block-diagonal structures, and proves a central theorem that subdirect products of just infinite groups with at most two virtually abelian factors are virtually diagonal by blocks. Key contributions include an equivalence between finite generation and containment of block subgroups, a comprehensive subdirect-product structure theorem, and corollaries proving subgroup separability (LERF) for important branch groups under the congruence subgroup property. The results bridge branch-group theory with subdirect-product rigidity, providing a powerful toolkit for understanding subgroup structure and enabling LERF results with potential for further rigidity applications.
Abstract
The aim of this paper is to describe the structure of the finitely generated subgroups of a family of branch groups, which includes the first Grigorchuk group and the Gupta-Sidki 3-group. This description is made via the notion of block subgroup. We then use this to show that all groups in the above family are subgroup separable (LERF). These results are obtained as a corollary of a more general structural statement on subdirect products of just infinite groups.