Virtual rational Betti numbers of nilpotent by abelian groups
In this paper we study virtual rational Betti numbers of a nilpotent by abelian group $G$ where the abelianization $N/N' of its nilpotent part $N$ satisfies certain tameness property. More precisely, we prove that if $N/N' is $2(c(n 1) 1) tame as a $G/N$ module, $c$ the nilpotency class of $N$ then \mathrm{vb}_j(G)\sup_{M\in\mathcal{A}_G}\dim_\mathbb{Q} H_j(M,\mathbb{Q} is finite for all $0\leq j\leq n$ where \mathcal{A}_G$ is the set of all finite index subgroups of $G$