arithmetic Kleinian groups from quaternion algebras gives rise to particularly interesting hyperbolic manifolds. On the other hand, they are in some sense "rare" among hyperbolic 3-manifolds (for example hyperbolic Dehn surgery on a fixed manifold results in a non-arithmetic manifold for almost all parameters). The hyperbolisation theorem In contrast to the explicit constructions above it is possible to deduce the existence of a complete hyperbolic structure on a 3–manifold purely from topological information. This is a consequence of the Geometrisation conjecture and can be stated as follows (a statement sometimes referred to as the "hyperbolisation theorem", which was proven by Thurston in the special case of Haken manifolds): A particular case is that of a surface bundle over the circle: such manifolds are always irreducible, and they carry a complete hyperbolic metric if and only if the monodromy is a pseudo-Anosov map. Another consequence of the Geometrisation conjecture is that any closed 3-manifold which admits a Riemannian metric with negative sectional curvatures admits in fact a Riemannian metric with constant sectional curvature -1. This is not true in higher dimensions. Virtual properties The topological properties of 3–manifolds are sufficiently intricate that in many cases it is interesting to know that a property holds virtually for a class of manifolds, that is for any manifold in the class there exists a finite covering space of the manifold with the property. The virtual properties of hyperbolic 3–manifolds are the objects of a series of conjectures by Waldhausen and Thurston, which were recently all proven by Ian Agol following work of Jeremy Kahn, Vlad Markovic, Frédéric Haglund, Dani Wise and others. The first part of the conjectures were logically related to the virtually Haken conjecture. In order of strength they are: (the surface subgroup conjecture) The fundamental group of any hyperbolic manifold of finite volume contains a (non-free) surface group (the fundamental group of a closed surface). (the Virtually Haken conjecture) Any hyperbolic 3–manifold of finite volume is virtually Haken; that is, it contains an embedded closed surface such that the embedding induces an injective map between fundamental groups. Any hyperbolic 3–manifold of finite volume has a finite cover with a nonzero first Betti number. Any hyperbolic 3–manifold of finite volume has a finite cover whose fundamental group surjects onto a non-abelian free group (such groups are usually called large). Another conjecture (also proven by Agol) which implies 1-3 above but a priori