The Godbillon measure of amenable foliations. (English) Zbl 0605.57015

The author obtains some sufficient condition for certain characteristic classes of foliations to vanish. Among other things, he proves the following: Let \({\mathcal F}\) be a codimension-n, \(C^ 2\)-foliation of a closed manifold. Suppose almost every leaf of \({\mathcal F}\) has subexponential growth. Then all the generalized Godbillon-Vey classes \(\Delta_*(y_ 1y_ Ic_ J)\) of F are zero. For the proof, the author studies measurable 1-cocycles of a groupoid \(\Gamma\), where \(\Gamma\) is roughly the groupoid of germs of local diffeomorphisms of sufficient transversals to \({\mathcal F}\), which are obtained by sliding points along the plaques.
The Godbillon measure of \({\mathcal F}\), which was first introduced by Duminy and generalized to the Weil measure by Heitsch and Hurder, is canonically obtained from the Radon-Nikodym cocycle \(d\nu\) ; \(d\nu ([\gamma]_ x)=\log \det (J\gamma)_ x\), \(J\gamma =\) Jacobian matrix of the local diffeomorphism \(\gamma\). If \(d\nu\) is cohomologous to zero, then the Godbillon measure is zero and so are the characteristic classes under consideration. The main part of the proof is, under the assumption of subexponential growth of leaves, to construct a family of cocycles \(a_{\epsilon}\) \((\epsilon >0)\) on \(\Gamma\), each of which is cohomologous to \(d\nu\) and \(\epsilon\)-tempered. This implies the nullity of the Godbillon measure.
Reviewer: T.Mizutani


57R30 Foliations in differential topology; geometric theory
57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
43A07 Means on groups, semigroups, etc.; amenable groups
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