A vanishing theorem for the tangential de Rham cohomology of a foliation with amenable fundamental groupoid (Q1430027)

The main result of the paper states that if \((X,\mathbb F)\) is a compact foliated topological space which is measurable and which is endowed with a leafwise Riemannian metric with non-positive curvature along the leaves such that all leaves are uniformly of rank at most \(r\), and if \(\mathcal F\) has an amenable fundamental groupoid, then the tangential de Rham cohomology groups vanish in degrees above \(r\). The first half of the paper sets up the terminology while the second proves the theorem and addresses a weakening of the hypotheses.