Agrarian and $L^2$-invariants

Henneke, Fabian; Kielak, Dawid

doi:10.48550/arxiv.1809.08470

Cited by 1 publication

(6 citation statements)

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“…The second author introduced the notion of an agrarian group in [Kie18]. In [HK19], the authors then developed a theory of algebraic invariants of nice spaces with an action of an agrarian group, which proceeds in analogy to the construction of L 2 -invariants. In this section, we will review the constructions and properties of these invariants, namely agrarian Betti numbers, agrarian torsion and agrarian polytopes, inasmuch as they are relevant to the proofs of our main results.…”

Section: Agrarian Invariantsmentioning

confidence: 99%

“…In this section, we will review the constructions and properties of these invariants, namely agrarian Betti numbers, agrarian torsion and agrarian polytopes, inasmuch as they are relevant to the proofs of our main results. For a full introduction, which also contains comparisons to L 2 -invariants and a discussion of the dependence of agrarian invariants on the choice of an agrarian embedding, we refer the reader to [HK19]. We will mostly follow the presentation therein, but use a different approach to the definition of agrarian torsion that is better suited for our computational purposes.…”

Section: Agrarian Invariantsmentioning

confidence: 99%

“…If C * is now a finite free ZG-chain complex that is D-acyclic with respect to α, then the D-chain complex D ⊗ ZG C * will be contractible. In [HK19], the agrarian torsion ρ D (C * ) of such a chain complex C * together with a choice of a basis was defined as a non-commutative D × ab -valued Reidemeister torsion in the sense of [Coh73]. First, out of a chain contraction of C * ,an element of the reduced K-group K 1 (D) is constructed, which is then mapped to D × ab via a map induced by the Dieudonné determinant of D. For the details of this definition, we refer the reader to [HK19, Section 4].…”

Section: Agrarian Torsionmentioning

confidence: 99%

“…While the construction of agrarian torsion in [HK19] is well-suited for the comparison to L 2 -torsion, for our current purposes a slightly different way of computing agrarian torsion is more convenient.…”

Section: Agrarian Torsionmentioning

confidence: 99%

“…Both of these theorems are proven using the machinery of agrarian invariants, introduced by the authors in [HK19].…”

Section: Introductionmentioning

confidence: 99%

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The agrarian polytope of two‐generator one‐relator groups

Henneke

Kielak

2020

J. London Math. Soc.

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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]).

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Section: Agrarian Invariantsmentioning

confidence: 99%

Section: Agrarian Invariantsmentioning

confidence: 99%