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**.

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