Three-manifolds, Foliations and Circles, I

TL;DR

This paper explores special foliations in three-manifolds that combine fibrations over the circle with circle bundles over surfaces, revealing their structure, dynamics, and potential for geometrization, especially in hyperbolic cases.

Contribution

It introduces a new class of foliations in three-manifolds that exhibit hyperbolic dynamics and sphere-filling leaf limits, expanding understanding of 3-manifold structures.

Findings

All such foliations admit transverse pseudo-Anosov flows.

In hyperbolic cases, leaves limit to sphere-filling Peano curves.

Examples include hyperbolic 3-manifolds of various homological types.

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.