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

TL;DR

The paper develops a quantitative polynomial cohomology theory for discrete groups and proves it coincides with ordinary cohomology when filling functions are polynomially bounded. It proves that Betti numbers of nilpotent groups are invariant under mutually cobounded L^p-measure equivalence, and derives vanishing results for non-cocompact lattices in rank-1 Lie groups. A central contribution is the induction of cohomology via ME-couplings and a transfer operator that yields injectivity and surjectivity statements between polynomial and ordinary cohomology, under explicit integrability and growth hypotheses. The framework is then applied to virtually nilpotent groups and rank-1 lattices to obtain concrete cohomological invariants, bounds on L^p-cohomology, and properties such as [T_3] for certain lattices. Overall, the work connects coarse geometric notions (ME, L^p integrability) with fine cohomological data through a chain-level, polynomial-growth perspective, enabling new rigidity and vanishing results.

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.

Theorems & Definitions (67)

• Theorem 1
• Theorem A
• Definition 2
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Corollary 3
• Corollary 4
• Definition 1.1