Geometry and topology of $\mathbb {R}$-covered foliations

An $\mathbb {R}$-covered foliation is a special type of taut foliation on a $3$-manifold: one for which holonomy is defined for all transversals and all time. The universal cover of a manifold $M$ with such a foliation can be partially compactified by a cylinder at infinity, somewhat analogous to the sphere at infinity of a hyperbolic manifold. The action of $\pi _1(M)$ on this cylinder decomposes into a product by elements of $\text {Homeo}(S^1)\times \text {Homeo}(\mathbb {R})$. The action on the $S^1$ factor of this cylinder is rigid under deformations of the foliation through $\mathbb {R}$-covered foliations. Such a foliation admits a pair of transverse genuine laminations whose complementary regions are solid tori with finitely many boundary leaves, which can be blown down to give a transverse regulating pseudo-Anosov flow. These results all fit in an essential way into Thurston’s program to geometrize manifolds admitting taut foliations.