Two-generator one-relator groups and marked polytopes

The agrarian polytope of two-generator one-relator groups

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.

双生成元单关系群的 JSJ 分解与多面体

JSJ decompositions and polytopes for two-generator one-relator groups

本文建立秩为2阿贝尔化的双曲双生成元单关系群的$Z_{max}$-JSJ分解与Friedl–Tillmann多面体的直接联系，并导出二次时间算法等结构性质。

Groups and polytopes

In a series of papers the authors associated to an $L^2$-acyclic group $Γ$ an invariant $\mathcal{P}(Γ)$ that is a formal difference of polytopes in the vector space $H_1(Γ;\Bbb{R})$. This invariant is in particular defined for most 3-manifold groups, for most 2-generator 1-relator groups and for all free-by-cyclic groups. In most of the above cases the invariant can be viewed as an actual polytope. In this survey paper we will recall the definition of the polytope invariant $\mathcal{P}(Γ)$ and we state some of the main properties. We conclude with a list of open problems.

The strong Atiyah and Lück approximation conjectures for one-relator groups

It is shown that the strong Atiyah conjecture and the Lück approximation conjecture in the space of marked groups hold for locally indicable groups. In particular, this implies that one-relator groups satisfy both conjectures. We also show that the center conjecture, the independence conjecture and the strong eigenvalue conjecture hold for these groups. As a byproduct we prove that the group algebra of a locally indicable group over a field of characteristic zero has a Hughes-free epic division algebra and, in particular, it is embedded in a division algebra.

Agrarian and $L^2$-invariants

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 $1$, t…