Abstract commensurability and the Gupta--Sidki group

Abstract

We study the subgroup structure of the infinite torsion $p$-groups defined by Gupta and Sidki in 1983. In particular, following results of Grigorchuk and Wilson for the first Grigorchuk group, we show that all infinite finitely generated subgroups of the Gupta--Sidki 3-group $G$ are abstractly commensurable with $G$ or $G\times G$. As a consequence, we show that $G$ is subgroup separable and from this it follows that its membership problem is soluble. Along the way, we obtain a characterization of finite subgroups of $G$ and establish an analogue for the Grigorchuk group.