  We develop the theory of agrarian invariants, which are algebraic
counterparts to $L^2$-invariants. Specifically, we introduce the notions of
agrarian Betti numbers, agrarian acyclicity, agrarian torsion and agrarian
polytope for finite free $G$-CW complexes together with a fixed choice of a
ring homomorphism from the group ring $\mathbb{Z} G$ to a skew field. For the
particular choice of the Linnell skew field $\mathcal{D}(G)$, this approach
recovers most of the information encoded in the corresponding $L^2$-invariants.
As an application, we prove that for agrarian groups of deficiency $1$, the
agrarian polytope admits a marking of its vertices which controls the
Bieri-Neumann-Strebel invariant of the group, improving a result of the second
author and partially answering a question of Friedl-Tillmann. We also use the
technology developed here to prove the Friedl-Tillmann conjecture on polytopes
for two-generator one-relator groups; the proof forms the contents of another
article.
