Three-manifolds, Foliations and Circles, I

Abstract

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.

60 pages, 10 figures

References in corpus (1)

• Minimal stretch maps between hyperbolic surfaces

Cited by in corpus (12)

• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
• Minimal stretch maps between hyperbolic surfaces
• Ideal boundaries of pseudo-Anosov flows and uniform convergence groups, with connections and applications to large scale geometry
• Hitchin characters and geodesic laminations
• Fuchsian Groups, Circularly Ordered Groups, and Dense Invariant Laminations on the Circle
• $C^0$ stability of boundary actions and inequivalent Anosov flows
• Partially hyperbolic diffeomorphisms homotopic to the identity on 3-manifolds
• Useful branched surfaces which carry nothing
• Real places and torus bundles
• Left orderability, foliations, and transverse $({\pi }_1,\mathbb{R})$ structures for $3$-manifolds with sphere boundary
• Characterization of Fuchsian groups as laminar groups & the structure theorem of hyperbolic 2-orbifolds
• Harmonic functions on R-covered foliations and group actions on the circle