# Extending concepts of "geometry" so that all prime three manifolds become "geometric".

3D geometrization is stated in terms of Thurston's eight locally homogeneous Riemannian geometries for certain prime three manifolds which are the topological building blocks for all prime three manifolds. We put aside for the moment the closed primes which carry one of these geometries of the form ${\mathbb S}^3$ or ${\mathbb S}^2 \times{\mathbb R}$, leaving six to discuss further.

We recall these homogeneous Riemannian geometries, as Klein persuaded, are included in more general homogeneous geometries like projective geometry or any locally homogeneous geometry defined by a Lie group. These Lie geometries have pictorial dynamical aspects related to **developing** mappings and **holonomy** representations.

This picture was elegantly formulated by Charles Ehresmann, student of Elie Cartan, around 1950 using invariant concepts. These picturesque ideas were often prominent in Thurston's work on three manifolds, dynamics and foliations.

In this joint work with Alice Kwon, CUNY Phd '19, we extend this Cartan-Ehresmann concept further to the notion of a "finite Lie diagram" geometry. The motivation was to accommodate the topological gluing data and the various subsets of Thurston Lie groups needed to build each element of the remaining vast set of prime three manifolds not carrying one pure Thurston geometry.

Theorem: 1) Each prime three manifold, for a certain finite Lie diagram, has a specific non empty set of finite Lie diagram geometries. 2) The finite dimensional moduli space of such can be described by the Lie groups involved and by Teichmuller spaces of related orbifolds. 3) These finite Lie diagram geometries have developing mappings which go from the universal covers of the prime three manifolds to the upper half space on which the six Thurston Lie groups are simultaneously represented. 4) The developing mappings turn out to be **bijective** and the holonomy mappings of the fundamental groups are isomorphisms onto **discrete subgroups** of the pushout topological group of the relevant finite Lie diagram. 5) For non trivial diagrams this pushout topological group is quite challenging being an infinite dimensional "Lie group" of real analytic diffeomorphisms of upper half space.