A vanishing theorem for the tangential de Rham cohomology of a foliation with amenable fundamental groupoid Kevin Corlette, Luis Hern´ andez Lamoneda and Alessandra Iozzi April 5, 2002 0. Introduction In his paper [8] on bounded cohomology, Gromov made two observations which were the springboard for this paper. One is that the bounded cohomology of a topological space with amenable fundamental group vanishes, while the second is that the bounded cohomology of a compact manifold with negative sectional curvature surjects onto the ordinary cohomology (with real coefficients) in degrees two and above. The only examples to which these obser- vations can both be applied are trivial, but if one extends one’s vision a bit, then there are interesting situations in which variants of both these facts apply. In this paper, we shall replace manifolds of negative curvature with topological foliations whose leaves have negative curvature. The analogue of the fundamental group in this setting is called the fundamental groupoid. This is a category whose objects are the points of the foliated space, and whose morphisms are homotopy classes of paths contained in leaves. The basic result we will prove is that, if the fundamental groupoid is amenable (in the spirit of the definitions of Zimmer [18, 19, 21]), and the sectional curvature of the leaves is bounded away from zero, then a suitable version of the tangential cohomology of the foliation vanishes in degrees two and above. The tangential cohomology is here defined as the cohomology of a suitable complex of leafwise differential forms, with the leafwise exterior derivative as differential. We shall in fact prove a more general result (Theorem 3.2) by allowing the leaves of the foliation to be manifolds of nonpositive curvature, everywhere of rank at most r (in a suitably uniform sense, see Definition 3.1). Under this less stringent hypothesis, we show that the tangential cohomology of an foliation with amenable fundamental groupoid whose leaves have this property vanishes in degree r + 1 and above. Here, the notion of rank is somewhat different from any of the standard definitions. We define the rank associated with a nonzero tangent vector v to a nonpositively curved manifold to be the largest dimension of a subspace to the tangent space such that every plane contained in the subspace and containing v has sectional curvature zero. In the case of symmetric spaces, this coincides with any of the usual notions of rank. In the case in which the leaves have strict negative curvature it is possible to approach these results in a manner completely parallel to Gromov’s approach in [8], and there may well be some interest in such an approach. But since our interest is mainly in the vanishing theorem, we have chosen to take a more direct route, avoiding completely any mention of K.C. was supported by NSF grant DMS-9971727 during the course of this work. L.H.L was supported by CONACyT project 37558-E, and sabbatical grants 010189 of CONACyT and SAB2000-0160 of the Ministerio de Educacion, Cultura y Deporte. 1