Positivity of Novikov–Shubin invariants for groups of type F∞

Establish whether the Novikov–Shubin invariants of every group of type F∞ are positive (equivalently, whether all capacities are finite in every degree for such groups).

Background

Lott–Lück conjectured positivity of Novikov–Shubin invariants for all chain complexes of finitely generated free ZΓ-modules, but Grabowski produced a counterexample using a solvable group that is not of type F∞. This motivates a weaker, still open question restricted to groups admitting finite K(Γ,1).

The paper develops bootstrapping frameworks for cofinal-measurability and capacity bounds and shows positivity for large families (e.g., infinite elementary amenable groups of type FP∞), but the general case for type F∞ remains unresolved.