Agrarian and -Betti numbers of locally indicable groups, with a twist.

We prove that twisted -Betti numbers of locally indicable groups are equal to the usual -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 with base space having locally indicable fundamental group, and with a simply-connected fiber , the first formula bounds -Betti numbers of in terms of -Betti numbers of and usual Betti numbers of ; the second formula computes exactly in terms of the same data, provided that is a high-dimensional sphere. We also present an inequality between twisted Alexander and Thurston norms for free-by-cyclic groups and 3-manifolds.