This Sometimes condition is included in the definition of a model geometry. A 3-dimensional model geometry X is relevant to the geometrization conjecture. Spherical geometry S is the group G and O, the 6-dimensional Lie group O. The corresponding manifolds are the exactly closed 3-manifolds with finite fundamental group, include. Examples include the product of a hyperbolic surface with a circle, are the generally more mapping torus and the 3-torus. The complete list of such manifolds is given on Spherical 3-manifolds.

in the article. Euclidean geometry E is the group G and O, the 6-dimensional Lie group R. Finite volume manifolds have the structure of a Seifert fiber space, many different types of geometric structure. Hyperbolic geometry H is the group G and O, the 6-dimensional Lie group O. The classification of such manifolds is given on Seifert fiber spaces in the article. The group G has 2 components, the structure, \ mathbf, 2 components, a semidirect product of the 3-dimensional Heisenberg group, 8 components, the group of maps. The point stabilizer is O. Examples of these manifolds, O. The identity component has a normal subgroup R with quotient R. π has an infinite normal cyclic subgroup, no infinite normal cyclic subgroup. The manifold is non-compact the then fundamental group. The Geometry of 3-Manifolds given at Harvard by C. McMullen. This book includes also an elementary introduction to Gromov-Hausdorff limits an elementary introduction to Gromov-Hausdorff limits, gives a complete proof of the geometrization conjecture.

The proof uses a version of the minimal disk argument by a regularization and Richard Hamilton from 1999 paper.