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

TL;DR

The paper establishes a precise link between the vanishing of the top-dimensional $L^2$-Betti number $b^{(2)}_{n}(G)$ for a residually poly-$\mathbb{Z}$ group $G$ of finite type and the existence of a poly-$\mathbb{Z}$ quotient with kernel having smaller rational cohomological dimension. It innovates by using the Okun--Schreve inductive ring framework to replace Novikov-type methods, proving that $b^{(2)}_{n}(G)=0$ implies a dimension drop along a residually poly-$\mathbb{Z}$ chain: there exists $i$ with $\mathrm{cd}_\mathbb{Q}(N_i) < n$ for some normal subgroup $N_i$. This leads to a characterization: $b^{(2)}_{n}(G)=0$ iff such a normal subgroup exists with $G/N_i$ poly-$\mathbb{Z}$, and yields corollaries in low dimensions (e.g., dimension two) and coherence of group rings. The results extend to mod $p$ analogues and provide a cleaner, more general proof that does not rely on the Novikov ring framework, broadening understanding of how $L^2$-invariants govern the algebraic structure 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.

Theorems & Definitions (24)

• Theorem 1.1
• Corollary 1.2
• Remark 2.1
• Definition 2.2
• Theorem 2.3: JaikinZapirain2020THEUO
• Definition 2.4
• Definition 2.5
• Definition 3.1
• Remark 3.2
• Theorem 3.3: OkunSchreve_DawidSimplified