Quantitative polynomial cohomology and applications to ${\text{L}}^p$-measure equivalence

We introduce a quantitative version of polynomial cohomology for discrete groups and show that it coincides with usual group cohomology when combinatorial filling functions are polynomially bounded. As an application, we show that Betti numbers of nilpotent groups are invariant by mutually cobounded ${\text{L}}^p$-measure equivalence. We also use this to obtain new vanishing results for non-cocompact lattices in rank 1 simple Lie groups.