# Hyperbolic group

In group theory, more precisely in geometric group theory, a **hyperbolic group**, also known as a *word hyperbolic group* or *Gromov hyperbolic group*, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov (1987). The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory. In a very influential (over 1000 citations ^{[1]}) chapter 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.

Members in this class of groups are often called *elementary hyperbolic groups* (the terminology is adapted from that of actions on the hyperbolic plane).

Examples of such are the fundamental groups of closed surfaces of negative Euler characteristic. Indeed, these surfaces can be obtained as quotients of the hyperbolic plane, as implied by the Poincaré—Koebe Uniformisation theorem.

Another family of examples of cocompact Fuchsian groups is given by triangle groups: all but finitely many are hyperbolic.

A further generalisation is given by groups admitting a geometric action on a CAT(k) space.^{[3]} There exist examples which are not commensurable to any of the previous constructions (for instance groups acting geometrically on hyperbolic buildings).

Groups having presentations which satisfy small cancellation conditions are hyperbolic. This gives a source of examples which do not have a geometric origin as the ones given above. In fact one of the motivations for the initial development of hyperbolic groups was to give a more geometric interpretation of small cancellation.

In some sense, "most" finitely presented groups with large defining relations are hyperbolic. For a quantitative statement of what this means see Random group.

Interesting examples in this class include in particular non-uniform lattices in rank 1 semisimple Lie groups, for example fundamental groups of non-compact hyperbolic manifolds of finite volume. Non-examples are lattices in higher-rank Lie groups and mapping class groups.

A group is said to be acylindrically hyperbolic if it admits a non-elementary acylindrical action on a (*not necessarily proper*) Gromov-hyperbolic space. This notion includes mapping class groups via their actions on curve complexes. Lattices in higher-rank Lie groups are (still!) not acylindrically hyperbolic.

In another direction one can weaken the assumption about curvature in the examples above: a *CAT(0) group* is a group admitting a geometric action on a CAT(0) space. This includes Euclidean crystallographic groups and uniform lattices in higher-rank Lie groups.

It is not known whether there exists a hyperbolic group which is not CAT(0).^{[15]}