Virtual rational Betti numbers of nilpotent-by-abelian groups

TL;DR

This paper investigates the conditions under which the virtual rational Betti numbers of nilpotent-by-abelian groups are finite, focusing on the tameness property of the abelianization of the nilpotent part.

Contribution

It establishes a new finiteness criterion for virtual rational Betti numbers based on the tameness of the abelianization in nilpotent-by-abelian groups.

Findings

Finite virtual Betti numbers for certain degrees under tameness conditions

Tameness property ensures control over homology growth

Provides a link between algebraic properties and topological invariants

Abstract

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 $G$.