The Gromov norm and foliations

Abstract

We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying foliation, in the geometric topology, and can therefore be used to study the question of which foliations arise as geometric limits of isotopy classes of other foliations. We show that this norm is non-trivial -- i.e. it distinguishes certain taut foliations of a given hyperbolic 3-manifold. In fact, modulo some standard conjectures, this norm should precisely detect amongst taut foliations of closed atoroidal 3-manifolds, those whose universal covers branch in both directions. A homotopy-theoretic refinement gives even more information in certain cases; in particular, we show that a taut foliation whose leaf space in the universal cover branches in at most one direction cannot be the geometric limit of a sequence of isotopies of a fixed taut foliation whose leaf space in the universal cover branches in both directions. Our technology also lets us produce examples of taut foliations which cannot be made transverse to certain geodesic triangulations of hyperbolic 3-manifolds, even after passing to a finite cover. Finally, our norm can be extended to actions of fundamental groups of manifolds on order trees, where it has similar upper semi-continuity properties.