On the top-dimensional $L^2$-Betti number of residually poly-$\mathbb Z$ groups

Abstract

Let $G$ be a residually poly-$\mathbb Z$ group of finite type. We prove that $G$ admits a poly-$\mathbb Z$ quotient with kernel $N$ satisfying $\mathrm{cd}_{\mathbb Q}(N) < \mathbb{cd}_{\mathbb Q}(G)$ if and only if the top-dimensional $L^2$-Betti number of $G$ vanishes.