Agrarian and $$\ell ^2$$-Betti numbers of locally indicable groups, with a twist

Abstract

AbstractWe prove that twisted $$\ell ^2$$ ℓ 2 -Betti numbers of locally indicable groups are equal to the usual $$\ell ^2$$ ℓ 2 -Betti numbers rescaled by the dimension of the twisting representation; this answers a question of Lück for this class of groups. It also leads to two formulae: given a fibration E with base space B having locally indicable fundamental group, and with a simply-connected fiber F, the first formula bounds $$\ell ^2$$ ℓ 2 -Betti numbers $$b_i^{(2)}(E)$$ b i ( 2 ) ( E ) of E in terms of $$\ell ^2$$ ℓ 2 -Betti numbers of B and usual Betti numbers of F; the second formula computes $$b_i^{(2)}(E)$$ b i ( 2 ) ( E ) exactly in terms of the same data, provided that F is a high-dimensional sphere. We also present an inequality between twisted Alexander and Thurston norms for free-by-cyclic groups and 3-manifolds. The technical tools we use come from the theory of generalised agrarian invariants, whose study we initiate in this paper.