Improved algebraic fibrings

We show that a virtually residually finite rationally solvable (RFRS) group G of type FPₙ (Q) virtually algebraically fibres with kernel of type FPₙ (Q) if and only if the first n ² -Betti numbers of G vanish, that is, bₚ^ (2) (G) = 0 for 0 p n. This confirms a conjecture of Kielak. We also offer a variant of this result over other fields, in particular in positive characteristic. As an application of the main result, we show that amenable virtually RFRS groups of type FP (Q) are virtually Abelian. It then follows that if G is a virtually RFRS group of type FP (Q) such that Z G is Noetherian, then G is virtually Abelian. This confirms a conjecture of Baer for the class of virtually RFRS groups of type FP (Q), which includes (for instance) the class of virtually compact special groups.