Virtual rational Betti numbers of nilpotent-by-abelian groups

Behrooz Mirzaii and Fatemeh Y. Mokari

Vol. 283 (2016), No. 2, 381–403

DOI: 10.2140/pjm.2016.283.381

Abstract

We study the 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 $2\left(c\left(n-1\right)-1\right)$-tame as a $G∕N$-module, where $c$ is the nilpotency class of $N$, then

$${vb}_j\left(G\right):=\underset{M\in {\mathcal{A}}_G}{sup}{dim}_{\mathbb{Q}}{H}_j\left(M,\mathbb{Q}\right)$$

is finite for all $0\le j\le n$, where ${\mathcal{A}}_G$ is the set of all finite-index subgroups of $G$.

Keywords

virtual Betti numbers, homology of groups, nilpotent-by-abelian groups, nilpotent action

Mathematical Subject Classification 2010

Primary: 20J05, 20J06

Milestones

Received: 30 August 2015

Accepted: 18 February 2016

Published: 22 June 2016