Editing Low-dimensional topology (section)

===Hyperbolic 3-manifolds===
{{Main|Hyperbolic 3-manifold}}

A [[hyperbolic 3-manifold]] is a [[3-manifold]] equipped with a [[complete space|complete]] [[Riemannian metric]] of constant [[sectional curvature]] -1.  In other words, it is the quotient of three-dimensional [[hyperbolic space]] by a subgroup of hyperbolic isometries acting freely and [[Properly discontinuous action|properly discontinuously]].  See also [[Kleinian model]].

Its thick-thin decomposition has a thin part consisting of tubular neighborhoods of closed geodesics and/or ends that are the product of a Euclidean surface and the closed half-ray.  The manifold is of finite volume if and only if its thick part is compact.  In this case, the ends are of the form torus cross the closed half-ray and are called '''cusps'''. Knot complements are the most commonly studied cusped manifolds.