Euclidean decompositions of noncompact hyperbolic manifolds

In this paper, we introduce a method for dividing up a noncompact hyperbolic manifold of finite volume into canonical Euclidean pieces. The construction first arose in the setting of surfaces (see (The Euclidean structure, of course, is not complete.) The conformal structure underlying this Euclidean structure does not agree with the underlying hyperbolic structure, but the two conformal structures are probably not too distant (cf. Sullivan's theorem