Subgroups of constructible nilpotent--by--abelian groups and a generalisation of a result of Bieri-Newmann-Strebel

We prove a $\Sigma$-version of the Bieri-Newmann-Strebel's result that for a finitely presented group $G$ without free subgroups of rank two $\Sigma^1(G)^c$ has no antipodal points. More precisely we prove that for such a group $G$$$conv_{\leq 2} ( \Bbb{R}_{> 0} \Sigma^1(G)^c) \subseteq \Bbb{R}_{>0} \Sigma^2(G)^c$$If $G$ is a finitely generated nilpotent-by-abelian group we show$$conv_{\leq 2} (\Bbb{R}_{> 0} \Sigma^1(G)^c) \subseteq \Bbb{R}_{>0} \Sigma^2(G, \Bbb{Z})^c$$The latter result is used in constructing a counter example to a conjecture of H. Meinert about homological properties of subgroups of constructible nilpotent-by-abelian groups.