  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 L\"uck 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.
