Hereditarily just-infinite torsion groups with positive first ℓ²-Betti number

2024/01/09 by Kionke, Steffen, Schesler, Eduard
#20C05 #20E07 #20E18 #20E26 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2401.04542

Abstract

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

• Approximating the first L²-betti number of residually finite groups
• Finitely generated infinite torsion groups that are residually finite simple
• Asymptotic invariants of residually finite just infinite groups
• L²-Betti numbers and non-unitarizable groups without free subgroups
• Normal generation and ℓ²-betti numbers of groups