Hyperbolic 3-manifolds and surface automorphisms

Hyperbolic Structures on 3-manifolds, Ii: Surface Groups and 3-manifolds Which Fiber over the Circle

The main result (0.1) of this paper is that every atoroidal three-manifold that fibers over the circle has a hyperbolic structure. Consequently, every fibered three-manifold admits a geometric decomposition. The main tool for constructing hyperbolic structures on fibered three-manifolds is the double limit theorem (4.1), which is of interest for its own sake and lays out general conditions unde...

Horocycles in hyperbolic 3-manifolds

Arithmetic of hyperbolic 3-manifolds

This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the “arithmetic topology”. Given 3-dimensional manifold M fibering over the circle we introduce an algebraic number field K = Q( √ d), where d > 0 is an integer number (discriminant) uniquely determined by M . The idea is to relate geometry of M to the arithmetic of field K. On this way, we show that V o...

Tameness of Hyperbolic 3-manifolds

Marden conjectured that a hyperbolic 3-manifold M with finitely generated fundamental group is tame, i.e. it is homeomorphic to the interior of a compact manifold with boundary [42]. Since then, many consequences of this conjecture have been developed by Kleinian group theorists and 3-manifold topologists. We prove this conjecture in theorem 10.2, actually in slightly more generality for PNC ma...

Minimum-volume Hyperbolic 3-manifolds

The classification of small-volume hyperbolic 3-manifolds has been an active problem for many years, ever since Thurston suggested that volume was a measure of the complexity of a hyperbolic 3-manifold. Quite recently in [GMM3] the author along with David Gabai and Robert Meyerhoff used a geometrical construction called a Mom-n structure to tackle the classification problem, and succeeded in sh...