Averages and the $\ell^{q,1}$-cohomology of Heisenberg groups

TL;DR

This paper investigates the behavior of averages in the $ ext{ell}^1$ cohomology of Lie groups, proving their vanishing for abelian and Heisenberg groups, which leads to the conclusion that the $ ext{ell}^1$ cohomology itself vanishes in these cases.

Contribution

It establishes the vanishing of averages in the $ ext{ell}^1$ cohomology for abelian and Heisenberg groups, completing previous research and showing the cohomology vanishes.

Findings

Averages vanish for abelian groups

Averages vanish for Heisenberg groups

$ ext{ell}^1$ cohomology vanishes in these cases

Abstract

Averages are invariants defined on the $\ell^1$ cohomology of Lie groups. We prove that they vanish for abelian and Heisenberg groups. This result completes work by other authors and allows to show that the $\ell^1$ cohomology vanishes in these cases.