The Gromov norm and foliations (Q5930038)

The author introduces a norm \(\|([M],{\mathcal F})\|_{FG}\) on the homology of an \(n\)-dimensional manifold \(M\) equipped with a foliation \(\mathcal F\), which is a refinement of the Gromov norm \(\|[M]\|_G\). In the author's definition, one restricts the admissible chains representing a homology class to those which are transverse -- that is, each singular map \(\sigma :\Delta^i\to M\) in the support of an admissible chain must induce a standard foliation on \(\Delta^i\), one which is topologically conjugate to an affine foliation. This norm depends in an upper semi-continuous way on the underlying foliation, in the geometric topology. The tension between the geometry of the manifold \(M\) and the local affine structure determined by the foliation is used to show that the foliated norm differs from the Gromov norm in certain cases, in a manner which reflects the topology and the geometry of the foliation. In particular, the author proves the following results: Theorem. Let \({\mathcal F} \) be a foliation on \(M\) whose universal cover is topolog ically conjugate to the standard foliation of \(\mathbb{R}^n\) by the horizontal subspaces \(\mathbb{R}^{n-1}\). Then \[ \|[M]\|_G=\|([M],{\mathcal F})\|_{FG}. \] Theorem. Suppose that \(\mathcal F\) is a taut foliation with one-sided branching. Then \[ \|[M]\|_G=\|([M],{\mathcal F})\|_{FG}. \] Theorem. Suppose that \(M\) is hyperbolic and that \(\mathcal F\) is asymptotically sep arated. Then \[ \|[M]\|_G<\|([M],{\mathcal F})\|_{FG}. \] Here, \(\mathcal F\) is said to be asymptotically separated if for some leaf \(\lambda \) of \(\tilde{\mathcal F}\) there is a pair of open hemispheres \(H^+\), \(H^-\) in the complement of \(\lambda\) in the universal cover \(H^n\) which are separated by \(\lambda\). Using a homotopy-theoretic refinement, the author shows 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. The techniques allows him also to 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, the author extends his norm to actions of fundamental groups of manifolds on order trees, where it has similar upper semi-continuity properties. A corrected version of this paper was distributed with issue 1 of volume 11 of this journal.