A Vanishing Theorem for the Tangential de Rham Cohomology of a Foliation with Amenable Fundamental Groupoid

For a space X carrying a foliation , the authors study the cohomology of the complex of differential forms which are smooth along the leaves and transversally locally L. It is shown that, if the leaves are nonpositively curved manifolds of rank at least r in a suitable uniform sense and the fundamental groupoid of the foliation is amenable, then the cohomology vanishes in degrees above r. This result is inspired by some of Gromov's results on bounded cohomology.