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

TL;DR

This paper establishes a precise condition linking the vanishing of the top-dimensional $L^2$-Betti number of residually poly-$b Z$ groups to the existence of certain poly-$b Z$ quotients with kernels of smaller rational cohomological dimension.

Contribution

It proves an if and only if condition connecting the top-dimensional $L^2$-Betti number to the existence of specific poly-$b Z$ quotients with smaller rational cohomological dimension.

Findings

Top-dimensional $L^2$-Betti number vanishes under certain quotient conditions.

Characterization of residually poly-$b Z$ groups via $L^2$-Betti numbers.

Conditions for the existence of poly-$b Z$ quotients with smaller cohomological dimension.

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.