Hyperbolic Group

In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov in the early 1980s. He noticed that many results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface do not rely either on it having dimension two or even on being a manifold and hold in much more general context. In a very influential paper from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.