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

We prove that twisted \(\ell ^2\) ${\ell }^2$ -Betti numbers of locally indicable groups are equal to the usual \(\ell ^2\) ${\ell }^2$ -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 \(\ell ^2\) ${\ell }^2$ -Betti numbers \(b_i^{(2)}(E)\) $b_i^{(2)}(E)$ of E in terms of \(\ell ^2\) ${\ell }^2$ -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.