Abstract commensurability and the Gupta–Sidki group
Alejandra Garrido
Université de Genève, Switzerland
Abstract
We study the subgroup structure of the infinite torsion -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 are abstractly commensurable with or . As a consequence, we show that 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 and establish an analogue for the Grigorchuk group.
Cite this article
Alejandra Garrido, Abstract commensurability and the Gupta–Sidki group. Groups Geom. Dyn. 10 (2016), no. 2, pp. 523–543
DOI 10.4171/GGD/355