Ergodic theory and Weil measures for foliations.

*(English)*Zbl 0645.57021This paper relates the transverse measure theory of a foliation to its secondary characteristic classes. The first significant result of this type was the (unpublished) theorem of G. Duminy that foliations of codimension one without resilient leaves have vanishing Godbillon-Vey invariant. To prove this, Duminy localized the Godbillon-Vey invariant to a \(\sigma\)-additive, cohomology-valued measure on the \(\sigma\)-algebra of measurable, saturated sets. Under the hypothesis that no leaf is resilient, Duminy decomposed the manifold into a countable, disjoint union of such sets, on each of which the Godbillon-Vey measure vanished. This use of measure theory inspired a series of successively more sophisticated papers by Hurder and Heitsch-Hurder relating transverse measure theory and characteristic classes.

The paper under review has as its main result (Theorem 0.1) a vanishing theorem for residual characteristic classes of \(C^ 2\) foliations \({\mathcal F}\) of codimension n on compact manifolds without boundary. The result is that, if the measurable equivalence relation \({\mathcal R}({\mathcal F})\) on M, determined by the leaves of \({\mathcal F}\), is amenable, then all residual classes in degrees strictly greater than \(2n+1\) must vanish. This condition is invariant under the very weak relation of measurable orbit equivalence of foliations.

Without a further hypothesis, it need not be true that residual classes in degree \(2n+1\) also vanish (e.g., the Roussarie example and Thurston’s examples on \(S^ 3)\). A condition that implies amenability and also implies that the residual classes vanish in degree \(2n+1\) is that almost all leaves have subexponential growth (Corollary 0.2).

(Reviewer’s note. Growth terminology has not entirely standardized. As used here, subexponential growth of a leaf L means that its growth function \(g_ L\) satisfies \(\lim_{m\to \infty}(\log g_ L(M))/m=0\). It is not known whether the weaker nonexponential growth condition, \(\liminf_{m\to \infty}(\log g_ L(m))/m=0\), is sufficient for Corollary 0.2. In codimension one, it is sufficient by the result of Duminy.)

Other corollaries to the main theorem result from a reformulation of the amenability condition in terms of the von Neumann algebra \({\mathcal M}({\mathcal F})\). By R. Zimmer [Invent. Math. 41, 23-31 (1977; Zbl 0361.46061)], \({\mathcal R}({\mathcal F})\) is amenable if and only if \({\mathcal M}({\mathcal F})\) is approximately finite (i.e., the weak closure of an increasing sequence of finite dimensional subalgebras). Thus, if the foliation \({\mathcal F}\) is of codimension n and if \({\mathcal M}({\mathcal F})\) is approximately finite, all residual secondary classes in degrees greater than \(2n+1\) vanish (Corollary 0.3). In particular, if \({\mathcal M}({\mathcal F})\) has Murray-von Neumann type I, these classes vanish (Corollary 0.4).

Crucial to all of the above is a result in ergodic theory which the authors establish. This concerns cocycles \(\phi\) : \({\mathcal R}({\mathcal F})\to GL(m,{\mathbb{R}})\) that are tempered relative to a suitable metric d on \({\mathcal R}({\mathcal F})\). That is, there is a continuous function \(c: {\mathbb{R}}^+\to {\mathbb{R}}^+\) such that \(| \phi (x,y)| <c(d(x,y))\). The result (Theorem 0.5) is that, if \({\mathcal R}({\mathcal F})\) is ergodic and amenable, then there exists a maximal, amenable subgroup \(H\subset GL(m,{\mathbb{R}})\) and a tempered cocycle \(\theta\) : \({\mathcal R}({\mathcal F})\to H\) that is cohomologous to \(\phi\). Moreover, if the growth rates of the leaves of \({\mathcal F}\) are at most exponential of type a and the growth of \(\phi\) is at most exponential of type b, then, for all \(\epsilon >0\), one can choose \(\theta =\theta_{\epsilon}\) to have growth type at most \((4m-3)a+(8m-6)b+\epsilon.\)

The paper under review has as its main result (Theorem 0.1) a vanishing theorem for residual characteristic classes of \(C^ 2\) foliations \({\mathcal F}\) of codimension n on compact manifolds without boundary. The result is that, if the measurable equivalence relation \({\mathcal R}({\mathcal F})\) on M, determined by the leaves of \({\mathcal F}\), is amenable, then all residual classes in degrees strictly greater than \(2n+1\) must vanish. This condition is invariant under the very weak relation of measurable orbit equivalence of foliations.

Without a further hypothesis, it need not be true that residual classes in degree \(2n+1\) also vanish (e.g., the Roussarie example and Thurston’s examples on \(S^ 3)\). A condition that implies amenability and also implies that the residual classes vanish in degree \(2n+1\) is that almost all leaves have subexponential growth (Corollary 0.2).

(Reviewer’s note. Growth terminology has not entirely standardized. As used here, subexponential growth of a leaf L means that its growth function \(g_ L\) satisfies \(\lim_{m\to \infty}(\log g_ L(M))/m=0\). It is not known whether the weaker nonexponential growth condition, \(\liminf_{m\to \infty}(\log g_ L(m))/m=0\), is sufficient for Corollary 0.2. In codimension one, it is sufficient by the result of Duminy.)

Other corollaries to the main theorem result from a reformulation of the amenability condition in terms of the von Neumann algebra \({\mathcal M}({\mathcal F})\). By R. Zimmer [Invent. Math. 41, 23-31 (1977; Zbl 0361.46061)], \({\mathcal R}({\mathcal F})\) is amenable if and only if \({\mathcal M}({\mathcal F})\) is approximately finite (i.e., the weak closure of an increasing sequence of finite dimensional subalgebras). Thus, if the foliation \({\mathcal F}\) is of codimension n and if \({\mathcal M}({\mathcal F})\) is approximately finite, all residual secondary classes in degrees greater than \(2n+1\) vanish (Corollary 0.3). In particular, if \({\mathcal M}({\mathcal F})\) has Murray-von Neumann type I, these classes vanish (Corollary 0.4).

Crucial to all of the above is a result in ergodic theory which the authors establish. This concerns cocycles \(\phi\) : \({\mathcal R}({\mathcal F})\to GL(m,{\mathbb{R}})\) that are tempered relative to a suitable metric d on \({\mathcal R}({\mathcal F})\). That is, there is a continuous function \(c: {\mathbb{R}}^+\to {\mathbb{R}}^+\) such that \(| \phi (x,y)| <c(d(x,y))\). The result (Theorem 0.5) is that, if \({\mathcal R}({\mathcal F})\) is ergodic and amenable, then there exists a maximal, amenable subgroup \(H\subset GL(m,{\mathbb{R}})\) and a tempered cocycle \(\theta\) : \({\mathcal R}({\mathcal F})\to H\) that is cohomologous to \(\phi\). Moreover, if the growth rates of the leaves of \({\mathcal F}\) are at most exponential of type a and the growth of \(\phi\) is at most exponential of type b, then, for all \(\epsilon >0\), one can choose \(\theta =\theta_{\epsilon}\) to have growth type at most \((4m-3)a+(8m-6)b+\epsilon.\)

Reviewer: L.Conlon

##### MSC:

57R30 | Foliations in differential topology; geometric theory |

57R32 | Classifying spaces for foliations; Gelfand-Fuks cohomology |

28D99 | Measure-theoretic ergodic theory |