Hyperbolic group

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in the sense of Gromov, Gromov hyperbolic group

A group with a finite generating subset for which there is some constant such that

for all , where and is the smallest integer such that can be written as a product of elements in .

Hyperbolic groups are sometimes called word hyperbolic groups or negatively curved groups.

A finite group is a trivial example of a hyperbolic group. Free groups of finite rank (cf. Free group) and fundamental groups of compact Riemannian manifolds of negative sectional curvature (cf. Fundamental group; Riemannian manifold) are hyperbolic. Groups given by a finite presentation satisfying the small-cancellation condition are also hyperbolic. The free product of two hyperbolic groups is a hyperbolic group. If is a group and is a subgroup of finite index, then is hyperbolic if and only if is hyperbolic. Algebraic properties of hyperbolic groups can be obtained via geometric methods. Every hyperbolic group is finitely presented (cf. Finitely-presented group), has a solvable word problem and even a solvable conjugacy problem. If is a hyperbolic group with a fixed generating subset and if denotes the number of elements such that , then the growth function is rational (cf. Polynomial and exponential growth in groups and algebras). Every hyperbolic group is automatic in the sense of [a1]. A hyperbolic group is said to be elementary if it is finite or contains an infinite cyclic subgroup of finite index. Every non-elementary hyperbolic group contains a free subgroup of rank . Torsion-free hyperbolic groups (cf. Group without torsion) have finite cohomological dimension. It is not known (1996) whether every hyperbolic group admits a torsion-free subgroup of finite index.

See also Gromov hyperbolic space.


[a1] D.B.A. Epstein, J.W.W. Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson, W.P. Thurston, "Word processing in groups" , Bartlett and Jones (1992)
[a2] M. Coornaert, T. Delzant, A. Papadopoulos, "Géométrie et théorie des groupes: les groupes hyperboliques de Gromov" , Lecture Notes in Mathematics , 1441 , Springer (1991)
[a3] "Sur les groupes hyperboliques d'après Mikhael Gromov" E. Ghys (ed.) P. de la Harpe (ed.) , Progress in Maths. , 83 , Birkhäuser (1990)
[a4] M. Gromov, "Hyperbolic groups" S.M. Gersten (ed.) , Essays in Group Theory , MSRI Publ. , 8 , Springer (1987) pp. 75–263
How to Cite This Entry:
Hyperbolic group. M. Coornaert (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098