| In this thesis we examine the interactions between taut foliations and geometrical structures on 3-manifolds. By comparing the extrinsic and intrinsic geometry of leaves in the universal cover of a taut foliation, we can construct auxiliary topological structures on the manifold, associated to the foliation, and we can use these to derive topological and geometric information about the foliation and about the underlying manifold. For instance, for a foliation which is R -covered or has one-sided branching of an atoroidal 3-manifold, we produce a pair of very full genuine laminations---i.e. laminations with solid torus guts---transverse to such a foliation and to each other, which bind each leaf of the foliation, and come from a product structure on the universal cover which is rigid under perturbations of the underlying foliation. These laminations can be blown down to stable and unstable singular foliations of a pseudo-Anosov flow transverse to the foliation. This generalizes results of Thurston for manifolds which fiber or slither over S 1, and suggests a program for proving that the underlying manifold is hyperbolic.;There is also a constructive part of this thesis: we produce examples of nonuniform R -covered foliations and of foliations with one-sided branching on hyperbolic 3-manifolds, some of which can be chosen to be rational homology spheres. The existence of these foliations disproves a conjecture of Thurston. |
