In mathematics, a CAT(0) group is a finitely generated group with a group action on a CAT(0) space that is geometrically proper, cocompact, and isometric.
They form a possible notion of non-positively curved group in geometric group theory.
is said to be a CAT(0) group if there exists a metric space
such that: An group action on a metric space satisfying conditions 2 - 4 is sometimes called geometric.
is CAT(0) is replaced with Gromov-hyperbolicity of
However, contrarily to hyperbolicity, CAT(0)-ness of a space is not a quasi-isometry invariant, which makes the theory of CAT(0) groups a lot harder.
The suitable notion of properness for actions by isometries on metric spaces differs slightly from that of a properly discontinuous action in topology.
can be covered by finitely many balls
However, metric properness is a stronger condition in general.
The two notions coincide for proper metric spaces.
acts (geometrically) properly and cocompactly by isometries on a length space
is actually a proper geodesic space (see metric Hopf-Rinow theorem), and
is finitely generated (see Švarc-Milnor lemma).
In particular, CAT(0) groups are finitely generated, and the space
involved in the definition is actually proper.
be a group acting properly cocompactly by isometries on a CAT(0) space
This geometry-related article is a stub.