Groups that do not act by automorphisms of codimension-one foliations

Abstract

Let G be a finitely generated group having the property that any action of any finite-index subgroup of G by homeomorphisms of the circle must have a finite orbit. (By a theorem of E.Ghys, lattices in simple Lie groups of real rank at least two have this property.) Suppose that such a G acts on a compact manifold M by automorphisms of a codimension-one C2 foliation, F. We show that if F has a compact leaf, then some finite-index subgroup of G fixes a compact leaf of F. Furthermore, we give sufficient conditions for some finite-index subgroup of G to fix each leaf of F.