commensurability and the Gupta-Sidki group Alejandra Garrido AbstractWe 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 × 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.

We show that all GGS-groups with non-constant defining vector satisfy the congruence subgroup property. This provides, for every odd prime p, many examples of finitely generated, residually finite, non-torsion groups whose profinite completion is a pro-p group, and among them we find torsionfree groups. This answers a question of Barnea. On the other hand, we prove that the GGS-group with constant defining vector has an infinite congruence kernel and is not a branch group.

This article is an expanded version of the talks given by the authors at the Arbeitsgemeinschaft "Totally Disconnected Groups", held at Oberwolfach in October 2014. We recall the basic theory of automorphisms of trees and Tits' simplicity theorem, and present two constructions of tree groups via local actions with their basic properties: the universal group associated to a finite permutation group by M. Burger and S. Mozes, and the k-closures of a given group by C. Banks, M. Elder and G. Willis.

Finding the number of maximal subgroups of infinite index of a finitely generated group is a natural problem that has been solved for several classes of "geometric" groups (linear groups, hyperbolic groups, mapping class groups, etc). Here we provide a solution for a family of groups with a different geometric origin: groups of intermediate growth that act on rooted binary trees. In particular, we show that the non-torsion iterated monodromy groups of the tent map (a special case of some groups first introduced byŠunić in [32] as "siblings of the Grigorchuk group") have exactly countably many maximal subgroups of infinite index, and describe them up to conjugacy. This is in contrast to the torsion case (e.g. Grigorchuk group) where there are no maximal subgroups of infinite index. It is also in contrast to the above-mentioned geometric groups, where there are either none or uncountably many such subgroups.Along the way we show that all the groups defined byŠunić have the congruence subgroup property and are just infinite.

Abstract. We answer a question of Bartholdi, Siegenthaler and Zalesskii, showing that the congruence subgroup problem for branch groups is independent of the branch action on a tree. We prove that the congruence topology of a branch group is determined by the group; specifically, by its structure graph, an object first introduced by Wilson. We also give a more natural definition of this graph.