Asymptotic invariants of residually finite just infinite groups (Q6661100)

A group \(G\) is just infinite if it is infinite and all non-trivial normal subgroups of \(G\) are of finite index. The interest of this family of groups lies in the fact that, by Zorn's Lemma, every finitely generated infinite group has a just infinite quotient. If a just infinite group is not residually finite, then it is virtually a power of a simple group.\N\NThe second author and \textit{E. Schesler}, in [``Hereditarily just-infinite torsion groups with positive first \(\ell^{2}\)-Betti number'', Preprint, \url{arXiv:2401.04542}], constructed examples of finitely generated residually finite, hereditarily just infinite groups with positive first \(\ell^{2}\)-Betti number.\N\NIn contrast to the above result, the authors of the paper under review prove Theorem 1.1: Let \(\Gamma\) be a finitely generated residually-\(p\) just infinite group. Then \(b^{(2)}_{1}(\Gamma, \mathbb{Q})=0\).\N\NFurthermore, they prove that the normal homology rank gradient of a finitely generated, residually finite, just infinite group vanishes.