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

Abstract. A homotopy equivalence between a hyperbolic 3-manifold and a closed irreducible 3-manifold is homotopic to a homeomorphsim provided the hyperbolic manifold satisfies a purely geometric condition. There are no known examples of hyperbolic 3-manifolds which do not satisfy this condition. One of the central problems of 3-manifold topology is to determine when a homotopy equivalence between two closed orientable irreducible 3-manifolds is homotopic to a homeomorphism. If one of these manifolds is S 3, then this is Poincaré’s problem. The results of [Re], [Fr], [Ru], [Bo], and [HR] (see also [Ol]) completely solve this problem for maps between lens spaces. In particular there exist nonhomeomorphic but homotopy equivalent lens spaces (e.g. L(7,1) and L(7,2)), and there exist self-homotopy equivalences not homotopic to homeomorphisms (e.g. the self-homotopy equivalence of L(8,1) whose π1-map is multiplication by 3). By Waldhausen [W] (resp. Scott [S]) a homotopy equivalence between a closed Haken 3-manifold (resp. a Seifert-fibred space with infinite π1) and an irreducible 3-manifold can be homotoped to a homeomorphism. By Mostow [M] a homotopy equivalence between two closed hyperbolic 3-manifolds can be homotoped to a homeomorphism and in fact an isometry. However, the general case of homotopy equivalence between a hyperbolic 3-manifold and an irreducible 3-manifold remains to be investigated. These problems and results should be contrasted with the conjecture [T] that a closed irreducible orientable 3-manifold is either Haken, or Seifert fibred with infinite π1, or the quotient of S 3 by an orthogonal action, or the quotient of H 3 via a cocompact group of hyperbolic isometries. Theorem 1 [G2]. Let N be a closed, orientable, hyperbolic 3-manifold containing an embedded hyperbolic tube of radius (log 3)/2 =.549306... about a closed geodesic. Then: (i) If f: M → N is a homotopy equivalence where M is an irreducible 3-manifold, then f is homotopic to a homeomorphism. (ii) If f, g: M → N are homotopic homeomorphisms, then f is isotopic to g. (iii) The space of hyperbolic metrics on N is path connected.

The behaviour of finite covers of 3-manifolds is a very important, but poorly understood, topic. There are three, increasingly strong, conjectures in the field that have remained open for over twenty years – the virtually Haken conjecture, the positive virtual b1 conjecture and the virtually fibred conjecture. Any of these

3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65

Abstract. In this paper it is proven that if the group of covering translations of the covering space of a compact, connected, P 2-irreducible 3-manifold corresponding to a non-trivial, finitely-generated subgroup of its fundamental group is infinite, then either the covering space is almost compact or the subgroup is infinite cyclic and has normalizer a non-finitely-generated subgroup of the rational numbers. In the first case additional information is obtained which is then used to relate Thurston’s hyperbolization and virtual bundle conjectures to some algebraic conjectures about certain 3-manifold groups. 1.

Abstract. This paper develops the basic theory of conformal structures on finite subdivision rules. The work depends heavily on the use of expansion complexes, which are defined and discussed in detail. It is proved that a finite subdivision rule with bounded valence and mesh approaching 0 is conformal (in the combinatorial sense) if there is a conformal structure on the model subdivision complex with respect to which the subdivision map is conformal. This gives a new approach to the difficult combinatorial problem of determining when a finite subdivision rule is conformal. 1.

We introduce the notion of a topological geodesic in a 3-manifold. Under suitable hypotheses on the fundamental group, for instance word-hyperbolicity, topological geodesics are shown to have the useful properties of, and play the same role in several applications as, geodesics in negatively curved spaces. This permits us to obtain virtual rigidity results for 3-manifolds.

Let M be a one-cusped hyperbolic 3–manifold. A slope on the boundary of the compact core of M is called exceptional if the corresponding Dehn filling produces a non-hyperbolic manifold. We give new upper bounds for the distance between two exceptional slopes ˛ and ˇ in several situations. These include cases where M.ˇ/ is reducible and where M.˛ / has finite 1, or M.˛ / is very small, or M.˛ / admits a 1 –injective immersed torus.

Abstract We introduce the notion of a topological geodesic in a 3-manifold. Under suitable hypotheses on the fundamental group, for instance word-hyperbolicity, topological geodesics are shown to have the useful properties of, and play the same role in several applications as, geodesics in negatively curved spaces. This permits us to obtain virtual rigidity results for 3-manifolds. AMS Classification 57M10, 20F67; 57M50

Abstract. This paper gives an algebraic conjecture which is shown to be equivalent to Thurston’s Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincaré Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture, and the other is equivalent to the union of the Geometrization Conjecture and Thurston’s Virtual Bundle Conjecture. 1.