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

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 Følner sequence (such as polycylic groups) as well as some wreath products (such as arbitrary wreath products of finitely generated Abelian groups).

13 pages. Added references, clarified some calculations

References in corpus (2)

• Vanishing of the first reduced cohomology with values in an $L^p$-representation
• The asymptotic density of dead ends in non-amenable groups