Math 2.0

Group theory and generalizations Definition

CAT(0) group

Builds on Group actionIsometryDehn function

Definition

Let XX be a CAT(0) space, that is, a geodesic metric space with the following property. For every geodesic triangle Δ\Delta, let Δˉ\bar\Delta be a triangle in ℝ2\R^2 with the same side lengths. Then d(p,q)≤∥pˉ−qˉ∥d(p, q) \le \norm{\bar p - \bar q} for all points p,qp, q of Δ\Delta and their comparison points pˉ,qˉ\bar p, \bar q in Δˉ\bar\Delta. Let a group GG act on XX by isometries.

  1. The action is proper if each x∈Xx \in X has a radius r>0r > 0 such that {g∈G:gB(x,r)∩B(x,r)≠∅}\set{g \in G : g B(x, r) \cap B(x, r) \neq \emptyset} is finite.
  2. The action is cocompact if there is a compact set K⊆XK \subseteq X with GK=XGK = X.

The group GG is a CAT(0) group if it admits a proper and cocompact action by isometries on some CAT(0) space.

Remarks

Example: ℤn\Z^n acts on ℝn\R^n by translations. The Euclidean space ℝn\R^n is CAT(0), since its triangles are exactly as thin as their comparison triangles. The action is proper, because a ball of radius 1/31/3 meets only its own translate, and cocompact, with K=[0,1]nK = [0, 1]^n.

Example: a free group of finite rank acts properly and cocompactly on its Cayley graph for a free basis, which is a tree, and trees are CAT(0).

Non-example: the integer Heisenberg group is not a CAT(0) group. In a CAT(0) group every solvable subgroup is virtually abelian, and the Heisenberg group is nilpotent but not virtually abelian.

Such an action is also called geometric. Every CAT(0) group is finitely presented, and its Dehn function is at most quadratic.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Dehn function, concept, no evidence levelDehn functionconceptGroup action, concept, no evidence levelGroup actionconceptIsometry, concept, no evidence levelIsometryconceptCAT(0) group, definition, no evidence levelCAT(0) groupdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Dehn function concept
  2. Group action concept
  3. Isometry concept
  4. CAT(0) group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/CAT(0)_group
About this page
TagDXCTK6M7ZV
URIm2://DXCTK6M7ZV
Kinddefinition
WritingW0 stub
MSC20F67, 20F65
DefinesCAT(0) group
UsesGroup action, Isometry, Dehn function
Changed2 November 2026

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