Foliated norms on fundamental group and homology

Abstract

This paper compares two invariants of foliated manifolds which seem to measure the non-Hausdorffness of the leaf space: the transversal length on the fundamental group and the foliated Gromov norm on the homology. We consider foliations with the property that the set of singular simplices strongly transverse to the foliation satisfies a weakened version of the Kan extension property. (We prove that this assumption is fairly general: it holds for all fibration-covered foliations, in particular for all foliations of 3-manifolds without Reeb components.) For such foliations we show that vanishing of the transversal length implies triviality of the foliated Gromov norm, and, more generally, that uniform bounds on the transversal length imply explicit bounds for the foliated Gromov norm. This is somewhat surprising in view of the fact that transversal length is defined in terms of 1- and 2-dimensional objects.