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Geometry and topology of \(\mathbb{R}\)-covered foliations. (English) Zbl 0995.57005
Summary: 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.

MSC:
57M50 General geometric structures on low-dimensional manifolds
57R30 Foliations in differential topology; geometric theory
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M60 Group actions on manifolds and cell complexes in low dimensions
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References:
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