  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.
