Invariant measures for affine foliations

Abstract

A (transversely) affine foliation is a foliation with an atlas whose coordinate changes are locally affine. Such foliations arise naturally in the study of affine structures on manifolds. In this paper we prove that an affine foliation with nilpotent affine holonomy group always admits a nontrivial transverse measure. Two proofs are given: one for noncompact manifolds, and another, valid for compact manifolds with ( G , X ) (G,X) -foliations (not necessarily affine) having nilpotent holonomy. These results are applied to prove that a certain cohomology class on a compact affine manifold with nilpotent holonomy is nonzero. Examples are discussed.