Homology growth, hyperbolization, and fibering (Q6196903)

A remarkable result has been proved by \textit{I. Agol} ([Doc. Math. 18, 1045--1087 (2013; Zbl 1286.57019)]): All closed, hyperbolic 3-manifolds have a finite cover that fibers over a circle. In a geometric direction, \textit{G. Italiano} et al. [Invent. Math. 231, No. 1, 1--38 (2023; Zbl 1512.57039)], constructed the first finite volume hyperbolic 5-manifolds that fiber over a circle. In a more algebraic direction, \textit{D. Kielak} [J. Am. Math. Soc. 33, No. 2, 451--486 (2020; Zbl 1480.20102)] and \textit{S. P. Fisher} [``Improved algebraic fibrings'',Preprint, \url{arXiv:2112.00397}] showed that for a large class of groups, the existence of a virtual \(\mathbb{F}\)-homological fibering is controlled by vanishing of certain skew field Betti numbers. For \(\mathbb{F}=\mathbb{Q}\) these Betti numbers are the \(L^{2}\)-Betti numbers, and for general fields they have an interpretation as a measure of \(\mathbb{F}\)-homology growth of finite covers. In the paper under review, the authors use \(\mathbb{F}_{p}\)-homology growth (for odd \(p\)) to prove Theorem A: There exists a closed, odd-dimensional, aspherical manifold \(\mathcal{M}\) with word hyperbolic fundamental group that does not virtually fiber over a circle. In this paper, other interesting results are proven, which are too technical to report here.