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

R. Feres and D. Witte

Vol. 204 (2002), No. 1, 31–42

DOI: 10.2140/pjm.2002.204.31

Abstract

Let Γ be a finitely generated group having the property that any action of any finite-index subgroup of Γ by homeomorphisms of the circle must have a finite orbit. (By a theorem of É. Ghys, lattices in simple Lie groups of real rank at least 2 have this property.) Suppose that such a Γ 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 Γ fixes a compact leaf of ℱ. Furthermore, we give sufficient conditions for some finite-index subgroup of Γ to fix each leaf of ℱ.

Milestones

Received: 17 August 2000

Revised: 17 March 2001

Published: 1 May 2002