structure of a hyperbolic 3–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. Structure Manifolds of finite volume 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 part, where the injectivity radius is larger than an absolute constant; and its complement, the thin part, which is a disjoint union of solid tori and cusps. Geometrically finite manifolds 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. Manifolds with finitely generated fundamental group This is the larger class of hyperbolic 3–manifolds for which there is a satisfying structure theory. It rests on two theorems: The tameness theorem which states that such a manifold is homeomorphic to the interior of a compact manifold with boundary; The ending lamination theorem which provides a classification of hyperbolic structure on the interior of a compact manifold by its "end invariants". Construction of hyperbolic 3–manifolds of finite volume Hyperbolic polyhedra, reflection groups The oldest construction of hyperbolic manifolds, which dates back at least to Poincaré, goes as follows: start with a finite collection of 3-dimensional hyperbolic finite polytopes. Suppose that there is a side-pairing between the 2-dimensional faces of these polyhedra (i.e. each such face is paired with another, distinct, one so that they are isometric to each other as 2-dimensional hyperbolic polygons), and consider the space obtained by gluing the paired faces together (formally this is obtained as a quotient space). It carries a hyperbolic metric which is well-defined outside of the image of the 1-skeletons of the polyhedra. This metric extends to a hyperbolic metric