Coherent RFRS groups

Abstract

We prove that a finitely generated virtually RFRS group of cohomological dimension at most $2$ is coherent if and only if its second $L^{2}$-Betti number vanishes if and only if it is virtually free-by-cyclic. The non-vanishing of the second $L^{2}$-Betti number provides the first known global obstruction to coherence in any reasonably wide class of groups, allowing for proofs of incoherence without needing to exhibit explicit witnesses to incoherence. As applications of this result, we completely characterise coherence among two-dimensional Coxeter groups, confirming conjectures of Jankiewicz and Wise, and show that incoherence is generic in groups of nonpositive deficiency, confirming a conjecture of Wise. We also find that, among virtually compact special groups of virtual cohomological dimension two, coherence is algorithmically decidable and is a quasi-isometry, measure equivalence, and profinite invariant. In an appendix, Marco Linton applies one of the main results to prove that cubulated locally quasi-convex hyperbolic groups are virtually free-by-cyclic, solving problems of Abdenbi--Wise and Wise in the cubulated case.