Effective drilling and filling of tame hyperbolic 3-manifolds

Drilling long geodesics in hyperbolic 3-manifolds

Given a complete hyperbolic 3-manifold one often wants to compare the original metric to a complete hyperbolic metric on the complement of some simple closed geodesic in the manifold. In some cases this can be done by interpolating between the two metrics using hyperbolic cone-manifolds. We refer to such a deformation as drilling and results which compare the geometry of the original manifold t...

Dehn filling of cusped hyperbolic 3-manifolds with geodesic boundary

We define for each g > 2 and k > 0 a set Mg,k of orientable hyperbolic 3manifolds with k toric cusps and a connected totally geodesic boundary of genus g. Manifolds in Mg,k have Matveev complexity g+k and Heegaard genus g+1, and their homology, volume, and Turaev-Viro invariants depend only on g and k. In addition, they do not contain closed essential surfaces. The cardinality of Mg,k for a fix...

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