Isometry transformations of 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...

Horocycles in hyperbolic 3-manifolds

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...

Hyperbolic Geometry: Isometry Groups of Hyperbolic Space

The goal of this paper is twofold. First, it consists of an introduction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that preserve orientation, and deals with the topological properties of their discrete subgroups. In the...