On the cohomological dimension of kernels of maps to ℤ

Fisher, Sam P.

doi:10.2140/gt.2026.30.373

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On the cohomological dimension of kernels of maps to $\mathbb{Z}$

Sam P. Fisher

Geometry & Topology 30 (2026) 373–388

DOI: 10.2140/gt.2026.30.373

Abstract

For a group $G$ of type $FP (R)$ for a ring $R$ and a homomorphism $χ : G \to ℤ$ , we show that ${cd}_{R} (\ker χ) = {cd}_{R} (G) - 1$ if the top-dimensional cohomology of $G$ with coefficients in the Novikov rings ${\hat{R [G]}}^{\pm χ}$ vanishes. This criterion is applied to show that if $G$ is a finitely generated RFRS group of cohomological dimension $2$ , then $G$ is virtually free-by-cyclic if and only if $b_{2}^{(2)} (G) = 0$ . This answers a question of Wise, and generalises and gives a significantly shorter proof of a recent theorem of Kielak and Linton, where the same result is obtained under the additional hypotheses that $G$ is virtually compact special and hyperbolic. A consequence of the result is that all virtually RFRS groups of rational cohomological dimension $2$ with vanishing second $ℓ^{2}$ -Betti number are coherent. More generally, we show that if $G$ is a RFRS group of cohomological dimension $n$ and of type ${FP}_{n - 1} (ℚ)$ , then $G$ admits a virtual map to $ℤ$ with kernel of rational cohomological dimension $n - 1$ if and only if $b_{n}^{(2)} (G) = 0$ .

Keywords

free-by-cyclic groups, coherence, group cohomology, Novikov rings, L2 invariants

Mathematical Subject Classification

Primary: 20F65, 20J05

Secondary: 16S34

References

Publication

Received: 8 December 2024

Revised: 11 July 2025

Accepted: 10 August 2025

Published: 19 January 2026

Proposed: Mladen Bestvina

Seconded: Martin R Bridson, Roman Sauer

Authors

Sam P. Fisher
Mathematical Institute University of Oxford Oxford United Kingdom

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Volume 30, issue 1 (2026)