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Geometrization says `` any closed oriented three-manifold which is prime (nota connected sum) carries one of the eight Thurston geometries OR it hasincompressible torus walls whose complementary components each carry one offour particular Thurston geometries" (see Introduction and Figure 1). Thesegeometric components have finite volume for the hyperbolic geometries (the Hlabeled vertices). They also have finite volume for each of the two geometriesappearing as Seifert fibrations (the S labeled vertices). The remaining pieces(the I labeled vertices) have Euclidean geometries of linear volume growth.Then these vertex geometries are combined topologically to recover the originalmanifold. This, by cutting off the toroidal ends and then gluing the torusboundaries by affine mappings (indicated by the labeled edges in Figure 1). Thepoint of this work is to make the affine gluing respect an interpretation ofthe metric geometry in terms of a new notion of `` regional Lie generatedgeometry". The vertex regions use four geometries in Lie form combined in theoverlap edge regions via affine geometry. The Theorem solves, usingGeometrization, a 45 year old question/approach to the Poincar\'{e} Conjecture.This was described in a '76 Princeton Math dept. preprint and finallydocumented in the 1983 reference by Thurston and the second author.

Compl\'ement to the Thurston 3D-Geometrization