The agrarian polytope of two‐generator one‐relator groups

Abstract

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]).

Agrarian and $L^2$-invariants

“…We can also relate the thickness of the polytope in a given direction to another agrarian invariant, which in the case of two-generator one-relator groups will turn out to compute a measure of complexity for possible HNN splittings of the group. The proofs of these two facts are the content of [HK20] by the present authors.…”

Survey on L2‐invariants and 3‐manifolds

“…Theorem 3.7 shows that the Atyiah Conjecture is related to the question whether for a torsionfree group the group ring can be embedded into a skew field, see, for instance, [42].…”