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Abstract: The property that the polynomial cohomology with coefficients of a finitelygenerated discrete group is canonically isomorphic to the group cohomology iscalled the weak isocohomological property for the group. In the case when agroup is of type $HF^\infty$, i.e. that has a classifying space with thehomotopy type of a cellular complex with finitely many cells in each dimension,we show that the isocohomological property is equivalent to the universal coverof the classifying space satisfying polynomially bounded higher Dehn functions.If a group is hyperbolic relative to a collection of subgroups, each of whichis polynomially combable respectively $HF^\infty$ and isocohomological, thenwe show that the group itself has these respective properties too. Combiningwith the results of Connes-Moscovici and Dru{\c{t}}u-Sapir we conclude that agroup satisfies the Novikov conjecture if it is relatively hyperbolic tosubgroups that are of property RD, of type $HF^\infty$ and isocohomological.



Author: R. Ji, B. Ramsey

Source: https://arxiv.org/







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