Dimension drop in residual chains

We give a description of the Linnell division ring of a countable residually (poly- Z \mathbb {Z} virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group G G with coefficients in this Novikov ring implies the existence of a normal subgroup N ⩽ G N \leqslant G such that c d Q ( N ) > c d Q ( G ) cd_\mathbb {Q}(N) > cd_\mathbb {Q}(G) and G / N G/N is poly- Z \mathbb {Z} virtually nilpotent. As a consequence, we show that if G G is an RPVN group of finite type, then its top-degree ℓ 2 \ell ^2 -Betti number vanishes if and only if there is a poly- Z \mathbb {Z} virtually nilpotent quotient G / N G/N such that c d Q ( N ) > c d Q ( G ) cd_\mathbb {Q}(N) > cd_\mathbb {Q}(G)