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

Abstract

We prove that twisted ${\ell }^2$-Betti numbers of locally indicable groups are equal to the usual ${\ell }^2$-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 fibre $F$, the first formula bounds ${\ell }^2$-Betti numbers $b_i^{(2)}(E)$ of $E$ in terms of ${\ell }^2$-Betti numbers of $B$ and usual Betti numbers of $F$; the second formula computes $b_i^{(2)}(E)$ 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.

We fixed an error in the proof of Theorem 4.3 pointed out by Andrei Jaikin-Zapirain. The statement remains unchanged