Hyperbolic group

In group theory, a hyperbolic group, also called negatively curved group, word-hyperbolic group, Gromov-hyperbolic group, delta-hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry.

Related Topics:
Group theory - Group - Word metric - Hyperbolic geometry

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There are several equivalent definitions. The first is the so-called thin triangles condition, generally credited to Eliyahu Rips. Let G be a finitely generated group, and T be its Cayley graph with respect to a finite set of generators. By identifying each edge isometrically with the unit interval in mathbb R, we can define a metric on T by defining the distance between each pair of points x and y in T to be the minimum length over all paths from x to y. A shortest path between two points is called a geodesic segment.

Related Topics:
Eliyahu Rips - Cayley graph

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A triangle in T is simply three points (the vertices) with each pair being joined by a geodesic segment (a side). Let delta geq 0. A triangle is delta-thin if each side is contained in a delta-neighborhood of the other two sides. If every triangle in T is delta-thin, then we say G is delta-hyperbolic. This condition is actually a quasi-isometric invariant, so in particular, does not depend on the set of generators chosen (although the actual value for delta may change).

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By imposing this condition on geodesic metric spaces in general, we arrive at the more general notion of δ-hyperbolic space.

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