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. We show that this norm is non-trivial — i.e. it distinguishes certain taut foliations of a given hyperbolic 3-manifold.¶Using a homotopy-theoretic refinement, we show that a taut foliation whose leaf space branches in at most one direction cannot be the geometric limit of a sequence of isotopies of a fixed taut foliation whose leaf space 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.