Hyperbolic 3-manifold

In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries (a Kleinian group). In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries (a Kleinian group). Hyperbolic 3–manifolds of finite volume have a particular importance in 3–dimensional topology as follows from Thurston's geometrisation conjecture proved by Perelman. The study of Kleinian groups is also an important topic in geometric group theory. Hyperbolic geometry is the most rich and least understood of the eight geometries in dimension 3 (for example, for all other geometries it is not hard to give an explicit enumeration of the finite-volume manifolds with this geometry, while this is far from being the case for hyperbolic manifolds). After the proof of the Geometrisation conjecture, understanding the topological properties of hyperbolic 3–manifolds is thus a major goal of 3-dimensional topology. Recent breakthroughs of Kahn–Markovic, Wise, Agol and others have answered most long-standing open questions on the topic but there are still many less prominent ones which have not been solved. In dimension 2 almost all closed surfaces are hyperbolic (all but the sphere, projective plane, torus and Klein bottle). In dimension 3 this is far from true: there are many ways to construct infinitely many non-hyperbolic closed manifolds. On the other hand the heuristic statement that 'a generic 3–manifold tends to be hyperbolic' is verified in many contexts. For example, any knot which is not either a satellite knot or a torus knot is hyperbolic. Moreover almost all Dehn surgeries on a hyperbolic knot yield a hyperbolic manifold. A similar result is true of links (Thurston's hyperbolic Dehn surgery theorem), and since all 3–manifolds are obtained as surgeries on a link in the 3–sphere this gives a more precise sense to the informal statement. Another sense in which 'almost all' manifolds are hyperbolic in dimension 3 is that of random models. For example random Heegaard splittings of genus at least 2 are almost surely hyperbolic (when the complexity of the gluing map goes to infinity). The relevance of the hyperbolic geometry of a 3–manifold to its topology also comes from the Mostow rigidity theorem, which states that the hyperbolic structure of a hyperbolic '–manifold of finite volume is uniquely determined by its homotopy type. In particular geometric invariant such as the volume can be used to define new topological invariants. In this case one important tool to understand the geometry of a manifold is the thick-thin decomposition. It states that a hyperbolic 3–manifold of finite volume has a decomposition into two parts: The thick-thin decomposition is valid for all hyperbolic 3–manifolds, though in general the thin part is not as described above. A hyperbolic 3–manifold is said to be geometrically finite if it contains a convex submanifold (its convex core) onto which it retracts, and whose thick part is compact (note that all manifolds have a convex core, but in general it is not compact). The simplest case is when the manifold does not have 'cusps' (i.e. the fundamental group does not contain parabolic elements), in which case the manifold is geometrically finite if and only if it is the quotient of a closed, convex subset of hyperbolic space by a group acting cocompactly on this subset.