  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).
