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

Contents

References

Further reading

References

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

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, ISBN 3-540-52977-2