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

Abstract

Let r be a finitely generated group having the property that any action of any finite-index subgroup of r 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 2 have this property.) Suppose that such a r acts on a compact manifold M by automorphisms of a codimension-one C 2 foliation, F. We show that if F has a compact leaf, then some finite-index subgroup of r fixes a compact leaf of F. Furthermore, we give sufficient conditions for some finite-index subgroup of r to fix each leaf of F.