---
tag: XB9QTY77ZA
kind: concept
name: CAT(0) group
msc: [20F67, 20F65]
sources:
  - url: https://en.wikipedia.org/wiki/CAT(0)_group
same_as:
  wikidata: Q5008888
  wikipedia: CAT(0) group
---

## Summary

A CAT(0) group is a group that acts properly and cocompactly by isometries on a [[CAT(0) space|1FAF2VNA4G]].
Such a space is a geodesic space whose triangles are at least as thin as Euclidean triangles.
It is the group-theoretic form of nonpositive curvature.
Examples include $\Z^n$, free groups, fundamental groups of closed nonpositively curved Riemannian manifolds, [[right-angled Artin groups|6D1DV09D6P]] and [[Coxeter groups|G6FND7X69J]].
CAT(0) groups are finitely presented, have a solvable word problem, and have a [[Dehn function|Q4HGMRA9JQ]] at most quadratic.

## Explanations

### L1 human

Draw a triangle on a saddle surface: its sides bend inwards compared with a flat triangle of the same side lengths.
A CAT(0) space is a space where every triangle is at least this thin, never fatter than a flat one.
A CAT(0) group is a group of symmetries of such a space.
It moves points around in a discrete way, and the translates of one compact piece cover the whole space.
For example, $\Z^2$ acts on the flat plane by translations, and the unit square tiles the plane.