Asymptotic invariants of residually finite just infinite groups

TL;DR

This paper investigates the asymptotic invariants of residually finite just infinite groups, establishing new results about their $L^2$-Betti numbers and homology rank gradient, contrasting previous constructions.

Contribution

It proves that finitely generated residually-$p$ just infinite groups have trivial first $L^2$-Betti number and zero normal homology rank gradient, advancing understanding of their invariants.

Findings

Finitely generated residually-$p$ just infinite groups have trivial first $L^2$-Betti number.

The normal homology rank gradient of such groups vanishes.

Contrasts with previous examples of groups with positive first $L^2$-Betti number.

Abstract

Recently, Eduard Schesler and the second author constructed examples of finitely generated residually finite, hereditarily just infinite groups with positive first $L^2$-Betti number. In contrast to their result, we show that a finitely generated residually-$p$ just infinite group has trivial first $L^2$-Betti number. Moreover, we prove that the normal homology rank gradient of a finitely generated, residually finite, just infinite group vanishes.