Virtual fibring of manifolds and groups
Dawid Kielak
TL;DR
The survey integrates algebraic, geometric, and analytic approaches to fibring over the circle, tracing how Stallings–Farrell reductions reduce fibering questions to finiteness properties of kernels and to BNS invariant criteria via Novikov rings. It then develops virtual fibering through obstructions such as Euler characteristic and $L^2$-homology, and leverages the Linnell skew-field framework together with RFRS structure to connect finite-index phenomena with Novikov and $L^2$ data. Applications include dimension-dropping phenomena and coherence results, with explicit examples like $F_2\times F_2$ illustrating limits of coherence. In higher dimensions, the discussion extends to PD$^n$ groups and hyperbolic manifolds, presenting criteria that link vanishing $L^2$-invariants to virtual fibring and outlining audacious conjectures for odd-dimensional hyperbolic manifolds, supported by partial results and a structured program. Overall, the paper synthesizes a broad toolkit—Stallings–Farrell reductions, BNS/N ovikov methods, $L^2$-techniques, and RFRS theory—to elucidate when manifolds and groups virtually fiber and to chart a path toward higher-dimensional generalizations.
Abstract
The topic of this survey is the phenomenon of fibring over the circle for manifolds, and its group-theoretic twin, algebraic fibring. We will discuss the state of the art, and explain briefly some of the ideas behind the more recent developments, focusing on RFRS groups and manifolds with such fundamental groups. Then we will move on to a more speculative part, where many conjectures about fibring in higher dimensions will be given. The conjectures vary in their level of plausibility, but even the boldest of them might share the fate of Thurston's Virtually Fibred Conjecture, about which Thurston famously said: ``This dubious-sounding question seems to have a definite chance for a positive answer''.