# Word-hyperbolic group

From Groupprops

## Contents

## Definition

A **word-hyperbolic group** is a finitely generated group such that the following equivalent conditions hold:

- For any finite generating set for , there exists a value such that the word metric for is -hyperbolic, i.e., for any triangle, each side is contained in the union of the -neighborhoods of the other two sides.
- There exists a finite generating set for and a value such that the word metric for is -hyperbolic, i.e., for any triangle, each side is contained in the union of the -neighborhoods of the other two sides.
- The group is a finitely presented group and has a linear isoperimetric function.

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finitely generated free group | |FULL LIST, MORE INFO | |||

finite group | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

group satisfying a quadratic isoperimetric inequality | Automatic group|FULL LIST, MORE INFO | |||

group satisfying a polynomial isoperimetric inequality | Automatic group|FULL LIST, MORE INFO | |||

group with solvable word problem | Automatic group, Group with polynomial-time solvable word problem|FULL LIST, MORE INFO |