Agrarian and $L^2$-invariants

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 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 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.

Agrarian and $\ell^2$-Betti numbers of locally indicable groups, with a twist

“…The notion of agrarian groups was introduced in [Kie20a], but the idea dates back to Malcev [Mc48]. The notion of agrarian invariants was later introduced and studied by Henneke and the first author in [HK18].…”

“…Remark 3.1. In [HK18], the authors define an agrarian map to be a 1-dimensional representation G → GL 1 (D) over a skew field. As we will see in Examples 3.3 and 3.13, general finite-dimensional representations arise naturally in the study of twisted invariants.…”

“…As we will see in Examples 3.3 and 3.13, general finite-dimensional representations arise naturally in the study of twisted invariants. We therefore generalize the work of [HK18] and define an agrarian map to be a general finite-dimensional representation.…”

“…For the rest of this subsection suppose n = 1, i.e., we have a homomorphism σ : G → D × . In [HK18], Henneke and the first author introduce the following rationalization of the agrarian map σ. Let K = ker q.…”

“…Remark 3.10. In the case n = 1, [HK18] provides an alternative definition for the agrarian polytope, which we call the HK-polytope for σ for the moment. The HKpolytope for σ is defined using σ : G → Ore(D * G fab ), the HK-rationalization of σ, where D * G fab denotes the twisted group ring in Section 3.2.…”