Hyperbolic group

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in the sense of Gromov, Gromov hyperbolic group

A group $G$ with a finite generating subset $S$ for which there is some constant $\delta=\delta(G,S)\geq0$ such that

$$\langle x,y\rangle\geq\min(\langle x,z\rangle,\langle y,z\rangle)-\delta$$

for all $x,y,z\in G$, where $\langle x,y\rangle=(|x|+|y|-|x^{-1}y|)/2$ and $|x|$ is the smallest integer $k\geq0$ such that $x$ can be written as a product of $k$ elements in $S\cup S^{-1}$.

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 $C'(1/6)$ small-cancellation condition are also hyperbolic. The free product of two hyperbolic groups is a hyperbolic group. If $G$ is a group and $G_0\subset G$ is a subgroup of finite index, then $G$ is hyperbolic if and only if $G_0$ 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 $G$ is a hyperbolic group with a fixed generating subset $S$ and if $a_n$ denotes the number of elements $x\in G$ such that $|x|=n$, then the growth function $f(t)=\sum_{n\geq0}a_nt^n$ 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 $2$. 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. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Coornaert (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article