hyperbolic group

A finitely generated group G is hyperbolic if, for some finite set of generatorsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath A of G, the Cayley graphMathworldPlanetmath Γ(G,A), considered as a metric space with d(x,y) being the minimum number of edges one must traverse to get from x to y, is a hyperbolic metric space.

Hyperbolicity is a group-theoretic property. That is, if A and B are finite sets of generators of a group G and Γ(G,A) is a hyperbolic metric space, then Γ(G,B) is a hyperbolic metric space.

examples of hyperbolic groups include finite groupsMathworldPlanetmath and free groupsMathworldPlanetmath. If G is a finite group, then for any x,yG, we have that d(x,y)|G|. (See the entry Cayley graph of S3 (http://planetmath.org/CayleyGraphOfS_3) for a pictorial example.) If G is a free group, then its Cayley graph is a real tree.

Title hyperbolic group
Canonical name HyperbolicGroup
Date of creation 2013-03-22 17:11:43
Last modified on 2013-03-22 17:11:43
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 6
Author Wkbj79 (1863)
Entry type Definition
Classification msc 05C25
Classification msc 20F06
Classification msc 54E35
Synonym hyperbolicity
Related topic RealTree