Virtual rational Betti numbers of nilpotent-by-abelian groups

Abstract

In this paper we study virtual rational Betti numbers of a nilpotent-by-abelian group $G$, where the abelianization $N/N^{\prime }$ of its nilpotent part $N$ satisfies certain tameness property. More precisely, we prove that if $N/N^{\prime }$ 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 $0\le j\le n$, where ${\mathcal{A}}_G$ is the set of all finite index subgroups of $G$.

19 pages