CAT(0) group

In mathematics, a CAT(0) group is a finitely generated group that admits a proper, cocompact, isometric action on a CAT(0) space.

A CAT(0) space is a complete geodesic metric space in which every geodesic triangle is at least as thin as the corresponding comparison triangle in Euclidean space; this condition encodes a notion of global non‑positive curvature in the sense of Alexandrov.

Formal definition

Let $G$ be a group. $G$ is called a CAT(0) group if there exists a metric space $X$ such that

1. $X$ is a CAT(0) space;
2. $G$ acts on $X$ by isometries (i.e., there is a homomorphism $G\to \mathrm{Isom}(X)$);
3. The action is geometrically proper: for every point $x\in X$ there is a radius $r>0$ such that the set ${g\in G\mid B(x,r)\cap g\cdot B(x,r) eq\varnothing}$ is finite;
4. The action is cocompact: there exists a compact subset $K\subset X$ whose translates under $G$ cover $X$ (i.e., $X = G\cdot K = \bigcup_{g\in G} gK$).

An action satisfying conditions 2–4 is often described as geometric.

Consequences

• Because the action is proper and cocompact on a proper length space, the Švarc–Milnor lemma implies that $G$ is finitely generated and that the space $X$ is quasi‑isometric to the Cayley graph of $G$.
• CAT(0) groups form a natural class of non‑positively curved groups within geometric group theory, analogous to the class of Gromov‑hyperbolic groups, though CAT(0)‑ness is not a quasi‑isometry invariant.

Examples

• Free abelian groups $\mathbb{Z}^n$ act geometrically on Euclidean space $\mathbb{R}^n$, which is CAT(0).
• Fundamental groups of closed non‑positively curved manifolds (e.g., tori, flat manifolds, certain locally symmetric spaces) are CAT(0) groups.
• Many Coxeter groups, right‑angled Artin groups, and right‑angled Coxeter groups admit actions on CAT(0) cube complexes and are thus CAT(0) groups.
• Certain braid groups have been shown to act geometrically on CAT(0) complexes.

Significance

CAT(0) groups provide a bridge between algebraic properties of groups and the geometric/topological structure of spaces on which they act. Studying such groups yields insights into rigidity phenomena, subgroup structure, and the large‑scale geometry of groups.

Related concepts

• CAT(0) space – the underlying metric space with non‑positive curvature.
• Geometric group theory – the broader field investigating groups via actions on geometric spaces.
• Gromov‑hyperbolic groups – another class of groups defined via actions on hyperbolic spaces; comparison often highlights differences between curvature conditions.

References

• Bridson, M. R.; Haefliger, A. Metric Spaces of Non‑Positive Curvature (1999).
• Gromov, M. Hyperbolic groups (1987).
• Wikipedia contributors, “CAT(0) group,” Wikipedia, The Free Encyclopedia, accessed 2024.