Word-hyperbolic group

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Definition

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

  1. For any finite generating set S for G, there exists a value \delta > 0 such that the word metric for (G,S) is \delta-hyperbolic, i.e., for any triangle, each side is contained in the union of the \delta-neighborhoods of the other two sides.
  2. There exists a finite generating set S for G and a value \delta > 0 such that the word metric for (G,S) is \delta-hyperbolic, i.e., for any triangle, each side is contained in the union of the \delta-neighborhoods of the other two sides.
  3. 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 properties
VIEW 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