The isocohomological property, higher Dehn functions, and relatively hyperbolic groups

The property that the polynomial cohomology with coefficients of a finitely generated discrete group is canonically isomorphic to the group cohomology is called the (weak) isocohomological property for the group. In the case when a group is of type HF∞, i.e. that has a classifying space with the homotopy type of a polyhedral complex with finitely many cells in each dimension, we show that the isocohomological property is geometric and is equivalent to the property that the universal cover of the classifying space has polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable, respectively HF∞ and isocohomological, then we show that the group itself has these respective properties. Combining with the results of Connes–Moscovici and Druţu–Sapir we conclude that a group satisfies the strong Novikov conjecture if it is hyperbolic relative to subgroups which are of property RD, of type HF∞ and isocohomological.