Agrarian and $L^2$-invariants
Volume 255 / 2021
Abstract
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 content of another article.
