The projective interpretation of the eight 3-dimensional homogeneous geometries.

Let (X; G) be a 3-dimensional homogeneous geometry, where X is a simply connected Riemannian space with a maximal group G of isometries, acting transitively on X with compact point stabilizers. G is maximal means that no proper extension of G can act on the Riemannian space X in such a way. We recall the Theorem. (Thurston) Any 3-dimensional homogeneous geometry (X; G) above that admits a compact quotient is equivalent (equivariant) to one of the geometries (X; G = Isom X) where the space X is one of E Our purpose is to model these geometries (X; G) on the real projective sphere PS 3 (R) X, or, especially, in the projective and aane space P 3 (R), A 3 (R), respectively, by special groups G of collineations. So we get a uniied method to attack Thurston's geometrization conjecture. Thurston guesses that each 3-dimensional compact orbifold (manifold) can be splitted (if occur) along 2-dimensional spherical (S 2-) orbifolds (manifolds) or Euclidean (E 2-) orbifolds (manifolds) in a unique way (up to isotopy) so that each piece (after some closing up procedure!) admits a complete, nite volume, geometric structure X=?. Here (X; G) is one of the 8 geometries above, and ? is a discrete subgroup of G = Isom X. I have been intending to disprove this conjecture by constructing counter examples: orbifolds and manifolds which can not wear any of the 8 metrics above. This intention has not been succeeded yet but the machinery developed here will be exploited, may be, in a future proof of the Thurston conjecture (?), (see 3], 4], 5] for further information). The point of our method is that wecanèasily recognize whether a transformation group ? as a * Work supported by the Hung.