The rank of the fundamental group of certain hyperbolic 3-manifolds fibering over the circle

We determine the rank of the fundamental group of those hyperbolic 3-manifolds fibering over the circle whose monodromy is a sufficiently high power of a pseudo-Anosov map. Moreover, we show that any two generating sets with minimal cardinality are Nielsen equivalent.Comment: This is the version published by Geometry & Topology Monographs on 29 April 200

Hyperbolic cone-manifolds with large cone-angles

We prove that every closed oriented 3-manifold admits a hyperbolic cone-manifold structure with cone-angle arbitrarily close to 2pi.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper24.abs.htm

Thick hyperbolic 3-manifolds with bounded rank

We construct a geometric decomposition for the convex core of a thick hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in terms of rank and injectivity radius on the Heegaard genus of M and on the radius of any embedded ball in the convex core of M.Comment: 170 pages, 17 figure

A Cantor set with hyperbolic complement

We construct a Cantor set in S^3 whose complement admits a complete hyperbolic metric

Counting Curves in Hyperbolic Surfaces

Let $\Sigma$ be a hyperbolic surface. We study the set of curves on $\Sigma$ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary $\gamma_0$. For example, in the particular case that $\Sigma$ is a once-punctured torus, we prove that the cardinality of the set of curves of type $\gamma_0$ and of at most length $L$ is asymptotic to $L^2$ times a constant.Comment: 49 pages, 11 (mostly hand-drawn) figure