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 M 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 and acylindrical; 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.
The word problem as measured by Dehn functions; connection to isoperimetric properties of manifolds via the Filling Theorem; the isoperimetric spectrum; the clear demarkation of hyperbolic groups and non-positively curved groups (again).
The isolation of the class of hyperbolic groups: hyperbolic versus atoroidal, cf.
Theorems establishing the equivalence of hyperbolic and atoroidal properties in various contexts (3-manifolds, analytic manifolds, new results in 2003--2004).
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