# Hyperbolic group

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.