CAT(0) group

A CAT(0) group is a group acting properly discontinuously and cocompactly on a CAT(0) space. This definition relies on several key concepts:

• Group: A group is an algebraic structure consisting of a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions—closure, associativity, identity, and invertibility—are satisfied.

• Action of a group: A group G acts on a space X if there is a function from G × X to X satisfying certain axioms, essentially specifying how each element of the group transforms the space.

• Properly discontinuous action: An action is properly discontinuous if for every compact subset K of X, the set {g ∈ G: gK ∩ K ≠ ∅} is finite. This means that the group doesn't "bunch up" the space too much.

• Cocompact action: An action is cocompact if the quotient space X/G is compact. This means that the group acts "transitively enough" to cover the space effectively.

• CAT(0) space: A CAT(0) space is a geodesic metric space satisfying the CAT(0) inequality, a condition involving geodesic triangles in the space. Intuitively, it is a space that is "non-positively curved" in a generalized sense. Examples include Euclidean spaces, hyperbolic spaces, and certain trees.

Therefore, a CAT(0) group is a group whose action on a CAT(0) space fulfills both the properly discontinuous and cocompact conditions. This implies significant geometric and algebraic properties for the group itself. The study of CAT(0) groups is a significant area of geometric group theory. Their properties are closely linked to the geometry of the CAT(0) space on which they act. These groups often exhibit rich algebraic structure and are amenable to various techniques from geometric topology and group theory. Many important classes of groups fall under the umbrella of CAT(0) groups.