longitude for each boundary torus, i.e. simple closed curves that are generators for the fundamental group of the torus. Let denote the manifold obtained from M by filling in the i-th boundary torus with a solid torus using the slope where each pair and are coprime integers. We allow a to be which means we do not fill in that cusp, i.e. do the "empty" Dehn filling. So M = .

We equip the space H of finite volume hyperbolic 3-manifolds with the geometric topology.