Geometry on all prime Three Manifolds
The point of this work is to construct geometric structures on the oriented closed prime three manifolds that don't at present already have them. One knows these compound prime three manifolds, have canonically up to deformation from the identity, incompressible torus walls whose complementary components are diffeomorphic to "elemental" prime three manifolds carrying single Thurston geometries. This metric geometry is generalized here to Lie geometry meaning an open cover by special coordinate charts in a model space whose transition mappings are related by one of several finite dimensional Lie groups acting on the model space. The Lie group is allowed to vary in a constrained manner from region to region in the manifold. Our geometric version of torus wall crossing after fixing finitely many parameters is rigid using covering spaces, pushouts, and sliding flat toroidal cylinders isometrically together. A new concept, Compound Lie Geometry, describes abstractly what these constructions produce. These Compound Lie Geometries in our case consist of sheaves of germs of charts into the upper half space which are related by four Lie groups acting there. It is key that each germ of the structure can be analytically continued along any path, like solving a classical ODE in the complex plane. This defines for each Lie Geometrized manifold a developing map of its universal cover into upper half space.