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Abstract
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.
Received: 2000-08-31
Revised: 2001-09-11
Published Online: 2008-02-22
Published in Print: 2002-03-01
© Australian Academic Press