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[parent] hyperbolic group (Definition)

A finitely generated group $ G$ is hyperbolic if, for some finite set of generators $ A$ of $ G$, the Cayley graph $ \Gamma(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 $ \Gamma(G,A)$ is a hyperbolic metric space, then $ \Gamma(G,B)$ is a hyperbolic metric space.

Simple examples of hyperbolic groups include finite groups and free groups. If $ G$ is a finite group, then for any $ x,y \in G$, we have that $ d(x,y) \le \vert G\vert$. (See the entry Cayley graph of $ S_3$ for a pictorial example.) If $ G$ is a free group, then its Cayley graph is a real tree.



"hyperbolic group" is owned by Wkbj79.
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See Also: real tree

Other names:  hyperbolicity

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Cross-references: real tree, free groups, finite groups, group, property, hyperbolic metric space, edges, number, metric space, Cayley graph, generators, finite set, finitely generated group
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This is version 3 of hyperbolic group, born on 2007-06-03, modified 2007-06-04.
Object id is 9514, canonical name is HyperbolicGroup.
Accessed 874 times total.

Classification:
AMS MSC: 20F06 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Cancellation theory; application of van Kampen diagrams)
 05C25 (Combinatorics :: Graph theory :: Graphs and groups)
 54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability)
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