  Geometrization says `` any closed oriented three-manifold which is prime (not
a connected sum) carries one of the eight Thurston geometries OR it has
incompressible torus walls whose complementary components each carry one of
four particular Thurston geometries" (see Introduction and Figure 1). These
geometric components have finite volume for the hyperbolic geometries (the H
labeled vertices). They also have finite volume for each of the two geometries
appearing 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 original
manifold. This, by cutting off the toroidal ends and then gluing the torus
boundaries by affine mappings (indicated by the labeled edges in Figure 1). The
point of this work is to make the affine gluing respect an interpretation of
the metric geometry in terms of a new notion of `` regional Lie generated
geometry". The vertex regions use four geometries in Lie form combined in the
overlap edge regions via affine geometry. The Theorem solves, using
Geometrization, a 45 year old question/approach to the Poincar\'{e} Conjecture.
This was described in a '76 Princeton Math dept. preprint and finally
documented in the 1983 reference by Thurston and the second author.

Geometric Topology;Analysis of PDEs;Classical Analysis and ODEs;Differential Geometry