We establish a sufficient condition for some modules M over the
group algebra ℤ[G ] to be of homological type FP2, where
G is a finitely generated split extension of abelian groups. This generalizes a result of Bieri
and Strebel [R. Bieri and R. Strebel. Valuations and finitely presented
metabelian groups. Proc. London Math. Soc.
(3) 41 (1980), 439–464] when M is the trivial module ℤ and
it establishes a special case of [D. H. Kochloukova. A new characterisation of
m -tame groups over finitely generated abelian groups. J. London
Math. Soc. (2) 60 (1999), 802–816, Conjecture K. S. Brown. Cohomology
of groups (Springer-Verlag, 1982)].
We prove a ∑-version of the result of Bieri, Neumann and Strebel [R. Bieri, W. D. Neumann and R. Strebel. A geometric invariant of discrete groups. Invent. Math.90 (1987), 451–477] that for a finitely presented group G without free subgroups of rank 2 the set ∑1(G)c has no antipodal points. More precisely, we prove that for such a group G
We show that if G is a finitely generated nilpotent-by-abelian group then
The latter result is used in constructing a counter-example to a conjecture of Meinert [H. Meinert. Iterated HNN-decomposition of constructible nilpotent-by-abelian groups. Comm. Algebra23 (1995), 3155–3164] concerning the homological properties of subgroups of constructible nilpotent-by-abelian groups.
We show that limit groups are free-by-(torsion-free nilpotent) and have non-positive Euler characteristic. We prove that for any non-abelian limit group the Bieri–Neumann–Strebel–Renz Σ-invariants are the empty set.
Let s ⩾ 3 be a natural number and G be a subdirect product of non-abelian limit groups intersecting each factor non-trivially. We show that the homology groups of any subgroup of finite index in G, in dimension i ⩽ s and with coefficients in ℚ, are finite-dimensional if and only if the projection of G to the direct product of any s of the limit groups has finite index. The case s = 2 is a deep result of M. Bridson, J. Howie, C. F. Miller III and H. Short.
Let G be a (topologically) finitely generated metabelian pro-p group. We prove the conjecture of J. King  that for any natural number m the group G embeds in a metabelian pro-p group of type FPm over ℤp.
We extend a result in [J. R. J. Groves and D. Kochloukova. Embedding properties of metabelian Lie algebras and metabelian discrete groups. J. London Math. Soc. (2) 73 (2006), 475–492.] which showed that for each m every finitely generated metabelian group G embeds in a quotient of a metabelian group of homological type FPm and furthermore that G embeds in a metabelian group of type FP4. More precisely, we show that for a fixed m every finitely generated metabelian group G embeds in a metabelian group of type FPm.