On the cohomological dimension of kernels of maps to Z

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 b₂⁽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 FPₙ₋₁, then G admits a virtual map to Z with kernel of rational cohomological dimension $n-1$ if and only if bₙ⁽²⁾(G) = 0.