Quantitative polynomial cohomology and applications to $\textrm L^p$-measure equivalence

TL;DR

This paper develops a quantitative polynomial cohomology theory for discrete groups, demonstrating its equivalence to classical cohomology under certain conditions, and applies it to invariance of Betti numbers and vanishing results in Lie groups.

Contribution

It introduces a new quantitative polynomial cohomology framework and applies it to establish invariance and vanishing results in geometric group theory.

Findings

Betti numbers of nilpotent groups are invariant under certain measure equivalences.

Established equivalence between quantitative and classical polynomial cohomology under polynomial bounds.

Derived new vanishing results for lattices in rank 1 Lie groups.

Abstract

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 $\textrm L^p$-measure equivalence. We also use this to obtain new vanishing results for non-cocompact lattices in rank 1 simple Lie groups.