The geometry of $\mathbb{R}$–covered foliations

Abstract

We study $ℝ$–covered foliations of 3–manifolds from the point of view of their transverse geometry. For an $ℝ$–covered foliation in an atoroidal 3–manifold $M$, we show that $\tilde{M}$ can be partially compactified by a canonical cylinder $S_{univ}^1\times ℝ$ on which ${\pi }_1\left(M\right)$ acts by elements of $Homeo\left(S^1\right)\times Homeo\left(ℝ\right)$, where the $S^1$ factor is canonically identified with the circle at infinity of each leaf of $\widetilde{\mathcal{ℱ}}$. We construct a pair of very full genuine laminations ${\Lambda }^{\pm }$ transverse to each other and to $\mathcal{ℱ}$, which bind every leaf of $\mathcal{ℱ}$. This pair of laminations can be blown down to give a transverse regulating pseudo-Anosov flow for $\mathcal{ℱ}$, analogous to Thurston’s structure theorem for surface bundles over a circle with pseudo-Anosov monodromy.

A corollary of the existence of this structure is that the underlying manifold $M$ is homotopy rigid in the sense that a self-homeomorphism homotopic to the identity is isotopic to the identity. Furthermore, the product structures at infinity are rigid under deformations of the foliation $\mathcal{ℱ}$ through $ℝ$–covered foliations, in the sense that the representations of ${\pi }_1\left(M\right)$ in $Homeo\left({\left(S_{univ}^1\right)}_t\right)$ are all conjugate for a family parameterized by $t$. Another corollary is that the ambient manifold has word-hyperbolic fundamental group.

Finally we speculate on connections between these results and a program to prove the geometrization conjecture for tautly foliated 3–manifolds.