Verified Computations for Hyperbolic 3-Manifolds

Verified Computations for Hyperbolic 3-Manifolds

Recent progress in field of 3-manifold topology has confirmed that each 3-manifold can be decomposed in to pieces that admit a geometric structure modelled on the quotient of one of eight simply connected spaces (for further background see the references below). By most accounts, the most common, and yet least understood of these geometric structures is the hyperbolic structure. A manifold M ad...

Geometric models for hyperbolic 3-manifolds

In [Mi4,BrocCM1], Minsky, Brock and Canary gave a proof of Thurston’s Ending Lamination Conjecture for indecomposable hyperbolic 3-manifolds. In this paper, we offer another approach to this which was inspired by the original. Many of the key results are similar, though the overall logic is somewhat different. A possible advantage is that much of the relevant theory of “hierarchies” and other, ...

Horocycles in hyperbolic 3-manifolds

Coarse hyperbolic models for 3-manifolds

The tameness and ending lamination conjectures together tell us that a hyperbolic 3-manifold with finitely generated fundamental group is determined by its topology and a finite number of “end invariants”. In this paper we describe how some of this theory generalises to a much broader class of metrics. To simplify the discussion, we focus on the particular case where the 3-manifold is homotopy ...

A finiteness theorem for hyperbolic 3-manifolds

We prove that there are only finitely many closed hyperbolic 3-manifolds with injectivity radius and first eigenvalue of the Laplacian bounded below whose fundamental groups can be generated by a given number of elements.