Vanishing of $\ell^p$-cohomology and transportation cost

TL;DR

This paper proves that for a specific class of amenable groups called transport amenable, the reduced $ ext{ell}^p$-cohomology is trivial, linking cohomology vanishing to transportation cost properties.

Contribution

It introduces the class of transport amenable groups and establishes the vanishing of reduced $ ext{ell}^p$-cohomology for these groups, expanding understanding of cohomology in relation to group properties.

Findings

Reduced $ ext{ell}^p$-cohomology is trivial for transport amenable groups.

Transport amenable groups include polycyclic groups and certain wreath products.

The class is characterized by bounded transportation cost sequences converging to a left-invariant mean.

Abstract

In this paper, it is shown that the reduced $\ell^p$-cohomology is trivial for a class of finitely generated amenable groups called transport amenable. These groups are those for which there exist a sequence of measures $\xi_n$ converging to a left-invariant mean and such that the transport cost between $\xi_n$ displaced by multiplication on the right by a fixed element and $\xi_n$ is bounded in $n$. This class contains groups with controlled F{\o}lner sequence (such as polycylic groups) as well as some wreath products (such as arbitrary wreath products of finitely generated Abelian groups).