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We present a new method to construct finitely generated, residually finite, infinite torsion groups. In contrast to known constructions, a profinite perspective enables us to control finite quotients and normal subgroups of these torsion groups. As an application, we describe the first examples of residually finite, hereditarily just-infinite groups with positive first ℓ2-Betti-number. In addition, we show that these groups have polynomial normal subgroup growth, which answers a question of Barnea and Schlage-Puchta.

Related project(s):
58Profinite perspectives on l2-cohomology

We prove that every finitely generated, residually finite group G embeds into a finitely generated perfect branch group such that many properties of G are preserved under this embedding. Among those are the properties of being torsion, being amenable and not containing a non-abelian free group. As an application, we construct a finitely generated, non-amenable torsion branch group.

Related project(s):
58Profinite perspectives on l2-cohomology

We prove that the minimal representation dimension of a direct product G of non-abelian groups G1,…,Gn is bounded below by n+1 and thereby answer a question of Abért. If each Gi is moreover non-solvable, then this lower bound can be improved to be 2n. By combining this with results of Pyber, Segal, and Shusterman on the structure of boundedly generated groups we show that branch groups cannot be boundedly generated.

Related project(s):
58Profinite perspectives on l2-cohomology

We introduce the notion of a telescope of groups. Very roughly a telescope is a directed system of groups that contains various commuting images of some fixed group B. Telescopes are inspired from the theory of groups acting on rooted trees. Imitating known constructions of branch groups, we obtain a number of examples of B-telescopes and discuss several applications. We give examples of 2-generated infinite amenable simple groups. We show that every finitely generated residually finite (amenable) group embeds into a finitely generated (amenable) LEF simple group. We construct 2-generated frames in products of finite simple groups and show that there are Grothendieck pairs consisting of amenable groups and groups with property (τ). We give examples of automorphisms of finitely generated, residually finite, amenable groups that are not inner, but become inner in the profinite completion. We describe non-elementary amenable examples of finitely generated, residually finite groups all of whose finitely generated subnormal subgroups are direct factors.

Related project(s):
58Profinite perspectives on l2-cohomology