Foliation dynamics and leaf invariants (Q1083078)

The author studies the relationship of leaf invariants of a foliation with the growth type of leaves. Leaf invariants are the characteristic classes of the normal bundles of the foliation restricted to a leaf, where the bundle is considered as a flat bundle. The main result in this direction is Theorem 3. Let L be a leaf of a \(C^ 2\)-foliation \({\mathcal F}\). If there is a leaf invariant \(y\in H^ m(L)\) with \(m>1\) and \(y\neq 0\), then the linear holonomy group of L is not amenable. The non-amenability of the linear holonomy group of L is related to the growth types of leaves of the foliation via Theorem 1. If the linear holonomy group of L is not amenable, then there is a leaf L' in the closure of L such that for any Riemannian metric on the foliated manifold L' has exponential growth.