Equality of cofibrant and Gorenstein projective ZG-modules for all groups

Establish that for every group G, the class Cof(ZG) of Benson cofibrant ZG-modules—i.e., all ZG-modules M such that the diagonal ZG-module M ⊗Z B(G, Z) is projective—coincides with the class GProj(ZG) of Gorenstein projective ZG-modules.

Background

Benson’s cofibrant modules over a group algebra are defined using the module B(G, Z) of bounded functions and play a central role in relating algebraic and geometric properties of groups. It is known that every cofibrant module is Gorenstein projective, so Cof(ZG) is always contained in GProj(ZG).

The conjectured reverse inclusion, if true for all groups, would align the cofibrant and Gorenstein projective frameworks and reduce questions about Gorenstein projective modules to classical homological algebra. The paper proves the equality under several conditions (e.g., for LH3-groups with certain properties on k), but the general case over Z remains unresolved.