23/02/26 16:49:34.45 ZAlHQVD3.net
>>828
つづき
URLﾘﾝｸ(en.wikipedia.org)
Hyperbolization theorem
For Perelman's generalization of Thurston's geometrization theorem to all 3-manifolds, see Geometrization conjecture.
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.
Statement
One form of Thurston's geometrization theorem states: If M is a compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of M has a complete hyperbolic structure of finite volume.
The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique.
URLﾘﾝｸ(en.wikipedia.org)
Atoroidal
In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non-boundary parallel, incompressible torus, or it may be defined algebraically, as a subgroup
Z ｘ Z of its fundamental group that is not conjugate to a peripheral subgroup (i.e., the image of the map on fundamental group induced by an inclusion of a boundary component). The terminology is not standardized, and different authors require atoroidal 3-manifolds to satisfy certain additional restrictions. For instance:
略
A 3-manifold that is not atoroidal is called toroidal.
(引用終り)
以上