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.
be its Cayley graph with respect to some finite set
is endowed with its graph metric (in which edges are of length one and the distance between two vertices is the minimal number of edges in a path connecting them) which turns it into a length space.
A priori this definition depends on the choice of a finite generating set
That this is not the case follows from the two following facts: Thus we can legitimately speak of a finitely generated group
of finite index) is also hyperbolic, for example the infinite dihedral group.
Members in this class of groups are often called elementary hyperbolic groups (the terminology is adapted from that of actions on the hyperbolic plane).
which acts properly discontinuously on a locally finite tree (in this context this means exactly that the stabilizers in
has an invariant subtree on which it acts with compact quotient, and the Svarc—Milnor lemma.
Such groups are in fact virtually free (i.e. contain a finitely generated free subgroup of finite index), which gives another proof of their hyperbolicity.
: it acts on the tree given by the 1-skeleton of the associated tessellation of the hyperbolic plane and it has a finite index free subgroup (on two generators) of index 6 (for example the set of matrices in
Note an interesting feature of this example: it acts properly discontinuously on a hyperbolic space (the hyperbolic plane) but the action is not cocompact (and indeed
-hyperbolic space and hence the Svarc—Milnor lemma tells us that cocompact Fuchsian groups are hyperbolic.
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.
Generalising the example of closed surfaces, the fundamental groups of compact Riemannian manifolds with strictly negative sectional curvature are hyperbolic.
For example, cocompact lattices in the orthogonal or unitary group preserving a form of signature
A further generalisation is given by groups admitting a geometric action on a CAT(k) space, when
[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.
is infinite: for example every group is hyperbolic relatively to itself via its action on a single point!).
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.
This notion includes mapping class groups via their actions on curve complexes.
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.