Vanishing of lp-cohomology and transportation cost

In this paper, it is shown that the reduced l(p)-cohomology is trivial for a class of finitely generated amenable groups called transport amenable. These groups are those for which there exists 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 (uniformly in n). This class contains groups with controlled Folner sequences (such as polycyclic groups) as well as some wreath products (such as H' (sic) H where H is finitely generated Abelian and the 'lamp state' group H' is finitely generated amenable).