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 $\mathbb Z\times\mathbb Z$ 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.