Groups, Geometry, and Dynamics

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Volume 10, Issue 2, 2016, pp. 523–543
DOI: 10.4171/GGD/355

Published online: 2016-06-09

Abstract commensurability and the Gupta–Sidki group

Alejandra Garrido[1]

(1) Université de Genève, Switzerland

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 solvable.

Along the way, we obtain a characterization of finite subgroups of $G$ and establish an analogue for the Grigorchuk group.

Keywords: Abstractly commensurable, structure of finitely generated subgroups, subgroup separable (LERF), Gupta–Sidki groups

Garrido Alejandra: Abstract commensurability and the Gupta–Sidki group. Groups Geom. Dyn. 10 (2016), 523-543. doi: 10.4171/GGD/355