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.

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.We use the agrarian invariants to solve the torsion-free case of a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, for such groups, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-Lück-Tillmann. Finally, 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.

Relying on the theory of agrarian invariants introduced in previous work, we solve a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, when the groups are additionally torsion-free, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-Lück-Tillmann.2010 Mathematics Subject Classification. Primary: 20J05; Secondary: 12E15, 16S35, 20E06, 57Q10. 1 arXiv:1912.04650v1 [math.AT] 10 Dec 2019 2 FABIAN HENNEKE AND DAWID KIELAKbe written as an HNN extension with induced character ϕ. They proved their conjectures in [FT15] under the additional hypothesis that the group G is residually {torsion-free elementary amenable}; later the first conjecture was confirmed by Friedl-Lück [FL17] under the weaker assumption that G is torsion-free and satisfies the strong Atiyah conjecture.Here a complete resolution of the first conjecture is offered:Theorem 5.12. If G is a group admitting a nice (2, 1)-presentation π, then M π ⊂ H 1 (G; R) ∼ = R 2 is an invariant of G (up to translation). Moreover, if G is torsionfree then P π = P Dr (G) for any choice of an agrarian embedding ZG → D.The notation P Dr (G) stands for the agrarian polytope, as introduced in [HK19], defined over the rationalisation D r of a skew field D. In fact, P Dr (G) is an invariant defined for any torsion-free two-generator one-relator group G other than the free group on two generators, even if b 1 (G) = 1.The second conjecture is also confirmed, assuming that G is torsion-free:Theorem 6.4. Let G be a torsion-free two-generator one-relator group other than the free group on two generators. Then for every epimorphism ϕ : G → Z we haveHere, c(G, ϕ) stands for the splitting complexity, and c f (G, ϕ) for the free splitting complexity.Both of these theorems are proven using the machinery of agrarian invariants, introduced by the authors in [HK19].(After the first version of this article appeared, Jaikin-Zapirain and López-Álvarez [JZLÁ19] published a proof of the strong Atiyah conjecture for torsion-free one-relator groups. This provides an alternative proof of the torsion-free case of our results as remarked in [FL17, Remark 5.5] and [FLT16, Theorem 5.2]).

Building on recent work of Jaikin-Zapirain, we provide a homological criterion for a ring to be a pseudo-Sylvester domain, that is, to admit a division ring of fractions over which all stably full matrices become invertible. We use the criterion to study skew Laurent polynomial rings over free ideal rings (firs). As an application of our methods, we prove that crossed products of division rings with free-by-{infinite cyclic} and surface groups are pseudo-Sylvester domains unconditionally and Sylvester domains if and only if they admit stably free cancellation. This relies on the recent proof of the Farrell–Jones conjecture for normally poly-free groups and extends previous results of Linnell–Lück and Jaikin-Zapirain on universal localizations and universal fields of fractions of such crossed products.

Building on recent work of Jaikin-Zapirain, we provide a homological criterion for a ring to be a pseudo-Sylvester domain, that is, to admit a non-commutative field of fractions over which all stably full matrices become invertible. We use the criterion to study skew Laurent polynomial rings over free ideal rings (firs).As an application of our methods, we prove that crossed products of division rings with free-by-cyclic and surface groups are pseudo-Sylvester domains unconditionally and Sylvester domains if and only if they admit stably free cancellation. This relies on the recent proof of the Farrell-Jones conjecture for poly-free groups and extends previous results of Linnell-Lück and Jaikin-Zapirain on universal localizations and universal fields of fractions of such crossed products. Contents 1.2. Ore and universal localizations 1.3. Weak and global dimensions 1.4. Stably freeness and stably finite rings 1.5. (Pseudo-)Sylvester domains 2. Towards Theorem A 2.1. Homological recognition principles for (pseudo-)Sylvester domains 2.2. The homological properties of D S and the proof of Theorem A 3. Applications: Theorem B and Theorem C 3.1. The Farrell-Jones conjecture and stably freeness 3.2. Examples and non-examples 3.3. Locally indicable groups and Hughes-freeness: identifying D E * G References