  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.
