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

TL;DR

This paper explores the relationship between the isocohomological property, higher Dehn functions, and relatively hyperbolic groups, establishing conditions under which these properties imply the Novikov conjecture for such groups.

Contribution

It proves the equivalence between the isocohomological property and polynomially bounded higher Dehn functions for groups of type HF∞, and shows how these properties extend to relatively hyperbolic groups.

Findings

Isocohomological property is equivalent to polynomially bounded higher Dehn functions for certain groups.

Relatively hyperbolic groups inherit properties like polynomial combability and HF∞ from their subgroups.

Groups with these properties satisfy the Novikov conjecture under specified conditions.

Abstract

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 $HF^\infty$, i.e. that has a classifying space with the homotopy type of a cellular complex with finitely many cells in each dimension, we show that the isocohomological property is equivalent to the universal cover of the classifying space satisfying polynomially bounded higher Dehn functions. If a group is hyperbolic relative to a collection of subgroups, each of which is polynomially combable (respectively $HF^\infty$ and isocohomological), then we show that the group itself has these respective properties too. Combining with the results of Connes-Moscovici and Dru{\c{t}}u-Sapir we conclude that a group satisfies the Novikov conjecture if it is relatively hyperbolic to subgroups that are of property RD, of type $HF^\infty$ and isocohomological.