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Three-manifolds, Foliations and Circles, I

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arxiv math/9712268 v1 pith:S6KPVJQR submitted 1997-12-30 math.GT math.DSmath.GR

William P. Thurston This is my paper · ORCID
classification math.GTmath.DSmath.GR
keywords circlefoliationscoverexampleshyperbolicmanifoldsthree-manifoldsuniversal
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This paper investigates certain foliations of three-manifolds that are hybrids of fibrations over the circle with foliated circle bundles over surfaces: a 3-manifold slithers around the circle when its universal cover fibers over the circle so that deck transformations are bundle automorphisms. Examples include hyperbolic 3-manifolds of every possible homological type. We show that all such foliations admit transverse pseudo-Anosov flows, and that in the universal cover of the hyperbolic cases, the leaves limit to sphere-filling Peano curves. The skew R-covered Anosov foliations of Sergio Fenley are examples. We hope later to use this structure for geometrization of slithered 3-manifolds.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension three

    math.DS 2025-05 conditional novelty 8.0 of 10

    Conservative partially hyperbolic diffeomorphisms homotopic to the identity on closed 3-manifolds with non-virtually-solvable fundamental group are always accessible and hence ergodic.

  2. Reconstructing flows from the orbit space

    math.DS 2025-09 conditional novelty 7.0 of 10

    A group action on a bifoliated plane with no infinite product regions comes from a pseudo-Anosov or expansive flow on a 3-manifold exactly when a certain space of leaf pairs admits a properly discontinuous, cocompact ...

  3. Transverse minimal foliations on unit tangent bundles and applications

    math.GT 2023-03 unverdicted novelty 5.0 of 10

    Transverse minimal foliations on unit tangent bundles of surfaces satisfy a dichotomy (Anosov intersection or Reeb surface), implying that volume-preserving partially hyperbolic diffeomorphisms on these spaces are ergodic.

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