Polynomial cohomology of groups was first introduced by Connes and Moscovici, and can be thought as a way of interpolating between bounded cohomology and usual group cohomology. Following work by Bader and Sauer, we introduce a quantitative version of polynomial cohomology and show that it coincides with group cohomology under polynomiality assumptions on filling functions. We give two applications of this result.
- We show that Betti numbers give an obstruction to L^p-measure equivalence of nilpotent groups for large values of p.
- We show vanishing of some cohomology spaces for non-uniform lattices in rank 1 simple Lie groups. In particular, we show that lattices acting on the octonionic hyperbolic plane enjoy a higher version of Kazhdan's property T.
This is based on joint work with Juan Paucar.