ON THE FP₃-CONJECTURE FOR METABELIAN GROUPS

THE $\Sigma^3$-CONJECTURE FOR METABELIAN GROUPS

“…One shows exactly as in [8] (by combinatorial homotopy arguments) that the composition j • h is injective. We next show that the image of this composition is the kernel of the boundary map Since M is finitely generated as a Q χ -module (we assumed that [χ] was in Σ M (Q)), it is actually of type FP ∞ .…”

“…The FP m -conjecture is true for m = 2 [11], for m = 3 in the split case [8], for G of finite Prüfer rank [1], and for the case when M is torsion and of Krull dimension 1 [13,15]. The direction (⇒) of the FP m -conjecture is true in the split case [15,16] (see also [22] for M torsion-free), and in the case when M is torsion [15,16].…”

“…In particular, the kernel of ∂ is a finitely generated Q χ -module. In [8], the diameter of a cell was defined to be the minimal radius of a ball in R n that contains the support of the cell. However, this notion is not suitable for working over the halfspace R n χ = {r ∈ R n | χ(r) > 0}.…”

“…In fact, the method of proof presented here is modeled on the techniques developed in paper [8] by Bieri and the first author and the idea of the diameter of cells introduced in [17]. The main point of [8] was the construction of a classifying K(M, 1)-complex for the Q-module M with Q-finite 3-skeleton (under the assumption that M is 3-tame), whereas here we construct a classifying K(M, 1)-complex for the Q χ -module M, where χ : Q −→ R is a non-trivial (that is, non-zero) character and Q χ is the positive submonoid of Q with respect to χ. In a sense one could say that the Σ m -conjecture is a monoid version of the FP m -conjecture.…”

Presentations of Inverse Semigroups, Their Kernels and Extensions

“…The relation rw = f (r, w) is a consequence of R ∪ U for all r ∈ Q 1 and w ∈ ((A ∪ A −1 ) ∪ (Q ∪ Q −1 )) * such that (r, w) (1, 1). [7] Presentations of kernels and extensions 295…”