Atoroidal

In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: a torus may be defined geometrically, as an embedded, non-boundary parallel, incompressible torus, or it may be defined algebraically, as a subgroup \mathbb Z\times\mathbb 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:

A 3-manifold that is not atoroidal is called toroidal.

References

  1. ↑ Apanasov, Boris N. (2000), Conformal Geometry of Discrete Groups and Manifolds, De Gruyter Expositions in Mathematics 32, Walter de Gruyter, p. 294, ISBN 9783110808056.
  2. ↑ Otal, Jean-Pierre (2001), The Hyperbolization Theorem for Fibered 3-manifolds, Contemporary Mathematics 7, American Mathematical Society, p. ix, ISBN 9780821821534.
  3. ↑ Chow, Bennett (2007), The Ricci Flow: Geometric aspects, Mathematical surveys and monographs, American Mathematical Society, p. 436, ISBN 9780821839461.
  4. ↑ Kapovich, Michael (2009), Hyperbolic Manifolds and Discrete Groups, Progress in Mathematics 183, Springer, p. 6, ISBN 9780817649135.
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