Agrarian and $L^2$-invariants

TL;DR

This paper introduces agrarian invariants as algebraic analogs of $L^2$-invariants, providing new tools for studying group properties and their geometric invariants, with applications to deficiency 1 groups and one-relator groups.

Contribution

It develops the theory of agrarian invariants, connecting algebraic and $L^2$-invariants, and applies this to group invariants and conjectures, advancing understanding in geometric group theory.

Findings

Agrarian polytope vertices can be marked to control the Bieri-Neumann-Strebel invariant.

The approach recovers most information of $L^2$-invariants via the Linnell skew field.

Proof of the Friedl-Tillmann conjecture for two-generator one-relator groups.

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 contents of another article.