Hyperbolic group

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

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.

A triangle in T is simply three points (the vertices) with each pair being joined by a geodesic segment (a side). Fix $\delta \geq 0$ . A triangle is $δ$ -thin if each side is contained in a $δ$ -neighborhood of the other two sides. If every triangle in T is $δ$ -thin, then we say G is $δ$ -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 $δ$ may change).

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

[edit] References

Mikhail Gromov, Hyperbolic groups. Essays in group theory, 75--263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.

[edit] Further reading

É. Ghys and P. de la Harpe (editors), Sur les groupes hyperboliques d'après Mikhael Gromov. Progress in Mathematics, 83. Birkhäuser Boston, Inc., Boston, MA, 1990. xii+285 pp. ISBN 0-8176-3508-4

Michel Coornaert, Thomas Delzant, Athanase Papadopoulos, "Géométrie et théorie des groupes : les groupes hyperboliques de Gromov", Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990, MR 92f:57003.