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 $mathrm{vb}_j(G):=sup_{Minmathcal{A}_G}dim_mathbb{Q} H_j(M,mathbb{Q})$ is finite for all $0leq jleq n$, where $mathcal{A}_G$ is the set of all finite index subgroups of $G$.