Agrarian and l2-Betti numbers of locally indicable groups,with a twist

We prove that twisted l(2)-Betti numbers of locally indicable groups are equal to theusual l(2)-Betti numbers rescaled by the dimension of the twisting representation; this answers a question of L & uuml;ck for this class of groups. It also leads to two formula e:given a fibration E with base space Bhaving locally indicable fundamental group, andwith a simply-connected fiber F, the first formula bounds l(2)-Betti numbersb(2)i(E)of Ein terms of l(2)-Betti numbers of B and usual Betti numbers of F; the secondformula computesb(2)i(E)exactly in terms of the same data, provided that F is ahigh-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.