---
tag: JWJ8YPR7EB
kind: concept
name: Hyperbolic group
msc: [20F67, 20F65]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperbolic_group
same_as:
  wikidata: Q81194
  wikipedia: Hyperbolic group
---

## Summary

A hyperbolic group is a [[finitely generated group|082DWWXMRJ]] whose [[Cayley graph|0GYQK9NDJF]] looks like a tree at large scale: its geodesic triangles are uniformly thin.
The notion was introduced by Gromov.
It covers free groups, fundamental groups of closed hyperbolic manifolds, and, in a precise sense, most finitely presented groups.
Hyperbolic groups are finitely presented, have a solvable word problem and conjugacy problem, and have a linear [[Dehn function|Q4HGMRA9JQ]].

## Explanations

### L1 human

In a tree, the three sides of a triangle meet in a tripod, so each side runs right along the other two.
In the hyperbolic plane the same is nearly true: triangles are thin, with every side close to the union of the other two.
A group is hyperbolic if its Cayley graph has this thin-triangle property, with a fixed bound on the thickness.
The free group on two generators is the simplest infinite example, since its Cayley graph is a tree.
The grid $\Z^2$ is not hyperbolic: its large triangles are fat, like triangles in the flat plane.