Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle

TL;DR

This paper explores the geometric structures of 3-manifolds that fiber over the circle, focusing on hyperbolic geometry, limit theorems for quasi-Fuchsian groups, and the quasi-isometric geometry of these manifolds, building on foundational work from the 1980s.

Contribution

It extends the understanding of hyperbolic structures on fibered 3-manifolds, including new results on limit behaviors of quasi-Fuchsian groups and their geometric properties.

Findings

Convergent subsequences of quasi-Fuchsian groups under certain conditions.

Analysis of the quasi-isometric geometry of quasi-Fuchsian 3-manifolds.

Application of the geometrization theorem to fibered 3-manifolds.

Abstract

Geometrization theorem, fibered case: Every three-manifold that fibers over the circle admits a geometric decomposition. Double limit theorem: for any sequence of quasi-Fuchsian groups whose controlling pair of conformal structures tends toward a pair of projectively measured laminations that bind the surface, there is a convergent subsequence. This preprint also analyzes the quasi-isometric geometry of quasi-Fuchsian 3-manifolds. This eprint is based on a 1986 preprint, which was refereed and accepted for publication, but which I neglected to correct and return. The referee's corrections have now been incorporated, but it is largely the same as the 1986 version (which was a significant revision of a 1981 version).