# Atoroidal

In mathematics, there are three definitions for atoroidal as applied to 3-manifolds:

• A 3-manifold is (geometrically) atoroidal if it does not contain an embedded, non-boundary parallel, incompressible torus.
• A 3-manifold is (geometrically) atoroidal if both of the following hold:
• It does not contain an embedded, non-boundary parallel, incompressible torus.
• It is acylindrical (also called anannular), meaning that it does not contain a properly embedded, non-boundary parallel, incompressible annulus.
• A 3-manifold is (algebraically) atoroidal if any subgroup of its fundamental group is conjugate to a peripheral subgroup, i.e. the image of the map on fundamental group induced by an inclusion of a boundary component.

Any algebraically atoroidal 3-manifold is geometrically atoroidal; but the converse is false. However, the mathematical literature often fails to distinguish between them, so one must ascertain any given author's intent.

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

### Other articles related to "atoroidal":

Hyperbolization Theorem - Statement
... One form of Thurston's geometrization theorem states If M is an compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of M has ... that the manifold M should be irreducible and atoroidal are necessary, as hyperbolic manifolds have these properties ... Thurston's hyperbolization conjecture states that a closed irreducible atoroidal 3-manifold with infinite fundamental group is hyperbolic, and this follows from Perelman's proof of the Thurston geometrization ...