PUB - Publikationen an der Universität Bielefeld

We develop the theory of agrarian invariants, which are algebraic coun-terparts 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 ZG to a skew field. For the particular choice of the Linnell skew field 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 poly-tope 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 content of another article.