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
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.