Thurston and many others developed the theory of geometrically finite ends of hyperbolic 3–manifolds. It remained to understand those ends which are not geometrically finite; such ends are called geometrically infinite.

This paper introduces a rigorous computer-assisted procedure for analyzing hyperbolic 3-manifolds. This procedure is used to complete the proof of several long-standing rigidity conjectures in 3-manifold theory as well as to

A proof that geometrically compressible one-sided splittings of non-Haken 3-manifolds are stabilised is given. This is a generalisation of a recent proof for the case of RP 3 and uses a modification of these techniques. Combined with known results about geometrically incompressible surfaces, the main result fully classifies one-sided splittings of small Seifert fibred spaces and the (6,1) Dehn filling of Figure 8 knot space. Drawing on minimal surface theory, it can also be used to show that non-Haken hyperbolic 3-manifolds have finitely many isotopy classes of one-sided splittings of bounded genus. 1

The central result of this paper, Theorem 6.1, gives a constraint that must be satisfied by the generators of any free, topologically tame Kleinian group without parabolic elements. The following result is case (a) of Theorem 6.1.

Abstract. We first summarize very briefly the topology of 3-manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization Conjecture for closed oriented 3-manifolds. The final two sections present evidence for the validity of these conjectures and outline an approach toward their proof.

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 ɛ-thin part of a hyperbolic manifold, for an arbitrary positive number ɛ, is defined to consist of all points through which there pass homotopically non-trivial curves of length at most ɛ. For small enough ɛ, the ɛ-thin part is geometrically very simple: it is a disjoint union of standard neighborhoods of closed geodesics and cusps. (Explicit descriptions of

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

The main theorem shows that if M is an irreducible compact connected orientable 3-manifold with nonempty boundary, then the classifying space BDiff (M rel ∂M) of the space of diffeomorphisms of M which restrict to the identity map on ∂M has the homotopy type of a finite aspherical CW-complex. This answers, for this class of manifolds, a question posed by M. Kontsevich. The main theorem follows from a more precise result, which asserts that for these manifolds the mapping class group H(M rel ∂M) is built up as a sequence of extensions of free abelian groups and subgroups of finite index in relative mapping class groups of compact connected surfaces.

Abstract. We describe a new approach to the well known canonical decompositions of 3-manifolds. 1.