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. Note: Examples 5.1 and 5.2 are incorrect (in particular $\mathbb{Z}\wr \mathbb{Z}$ and $(\mathbb{Z}/2\mathbb{Z})\wr {\mathbb{Z}}^2$ are not transport amenable); an erratum and addendum is appended to this version.

arXiv comment12 pages (original paper); followed by an erratum and addendum 12 pages. Examples 5.1 and 5.2 are incorrect

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