We give a description of the Linnell division ring of a countable residually (poly- Z 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 G such that c d Q (N) > c d Q (G) cd_ Q (N) > cd_ Q (G) and G / N G/N is poly- Z 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 ² -Betti number vanishes if and only if there is a poly- Z Z virtually nilpotent quotient G / N G/N such that c d Q (N) > c d Q (G) cd_ Q (N) > cd_ Q (G)
Fisher et al. (Tue,) studied this question.