# Commentarii Mathematici Helvetici

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**Volume 77, Issue 3, 2002, pp. 415–490**

**DOI: 10.1007/s00014-002-8348-9**

Foliations, topology and geometry of 3-manifolds: R-covered foliations and transverse pseudo-Anosov flows

Sérgio R. Fenley^{[1]}(1) Department of Mathematics, Florida State University, 208 James J. Love Building, FL 32306-4510, TALLAHASSEE, UNITED STATES

We analyse the topological and geometrical behavior of foliations on 3-manifolds. We consider the transverse structure of an R-covered foliation in a 3-manifold, where R-covered means that in the universal cover the leaf space of the foliation is Hausdorff. If the manifold is aspherical we prove that either there is an incompressible torus in the manifold; or there is a transverse pseudo-Anosov flow. It follows that manifolds with R-covered foliations satisfy the weak hyperbolization conjecture.

*Keywords: *Foliations, transverse flows, geometric structures on leaves, holonomy, transverse structure of foliation, universal circle, Gromov hyperbolic

Fenley Sérgio: Foliations, topology and geometry of 3-manifolds: R-covered foliations and transverse pseudo-Anosov flows. *Comment. Math. Helv.* 77 (2002), 415-490. doi: 10.1007/s00014-002-8348-9