On the cohomological dimension of kernels of maps to ℤ
Abstract
We prove 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 new 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. More generally, we show that if $G$ is a RFRS group of cohomological dimension $n$ and of type $\mathrm{FP}_{n-1}$, then $G$ admits a virtual map to $\mathbb Z$ with kernel of rational cohomological dimension $n-1$ if and only if $b_n^{(2)}(G) = 0$.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Homotopy and Cohomology in Algebraic Topology
Geometric and Algebraic Topology
Publication details
- Journal / source
- Geometry & Topology
- Publisher
- Mathematical Sciences Publishers
- Volume
- 30
- Issue
- 1
- Pages
- 373–388
- Publication date
- 19 January 2026
- Language
- EN