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 . 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 ) 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.
Definition.
Let formula_1 be a finitely generated group, and formula_2 be its Cayley graph with respect to some finite set formula_3 of generators. The set formula_2 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. The group formula_1 is then said to be "hyperbolic" if formula_2 is a hyperbolic space in the sense of Gromov. Shortly, this means that there exists a formula_7 such that any geodesic triangle in formula_2 is formula_9-thin, as illustrated in the figure on the right (the space is then said to be formula_9-hyperbolic).
A priori this definition depends on the choice of a finite generating set formula_3. That this is not the case follows from the two following facts:
Thus we can legitimately speak of a finitely generated group formula_1 being hyperbolic without referring to a generating set. On the other hand, a space which is quasi-isometric to a formula_9-hyperbolic space is itself formula_14-hyperbolic for some formula_15 but the latter depends on both the original formula_9 and on the quasi-isometry, thus it does not make sense to speak of formula_1 being formula_9-hyperbolic.
Remarks.
The Švarc–Milnor lemma states that if a group formula_1 acts properly discontinuously and with compact quotient (such an action is often called "geometric") on a proper length space formula_20, then it is finitely generated, and any Cayley graph for formula_1 is quasi-isometric to formula_20. Thus a group is (finitely generated and) hyperbolic if and only if it has a geometric action on a proper hyperbolic space.
If formula_23 is a subgroup with finite index (i.e., the set formula_24 is finite), then the inclusion induces a quasi-isometry on the vertices of any locally finite Cayley graph of formula_25 into any locally finite Cayley graph of formula_1. Thus formula_25 is hyperbolic if and only if formula_1 itself is. More generally, if two groups are commensurable, then one is hyperbolic if and only if the other is.
Examples.
Elementary hyperbolic groups.
The simplest examples of hyperbolic groups are finite groups (whose Cayley graphs are of finite diameter, hence formula_9-hyperbolic with formula_9 equal to this diameter).
Another simple example is given by the infinite cyclic group formula_31: the Cayley graph of formula_31 with respect to the generating set formula_33 is a line, so all triangles are line segments and the graph is formula_34-hyperbolic. It follows that any group which is virtually cyclic (contains a copy of formula_31 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).
Free groups and groups acting on trees.
Let formula_36 be a finite set and formula_37 be the free group with generating set formula_3. Then the Cayley graph of formula_37 with respect to formula_3 is a locally finite tree and hence a 0-hyperbolic space. Thus formula_37 is an hyperbolic group.
More generally we see that any group formula_1 which acts properly discontinuously on a locally finite tree (in this context this means exactly that the stabilizers in formula_1 of the vertices are finite) is hyperbolic. Indeed, this follows from the fact that formula_1 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.
An interesting example is the modular group formula_45: 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 formula_1 which reduce to the identity modulo 2 is such a group). 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 formula_1 is "not" quasi-isometric to the hyperbolic plane).
Fuchsian groups.
Generalising the example of the modular group a Fuchsian group is a group admitting a properly discontinuous action on the hyperbolic plane (equivalently, a discrete subgroup of formula_48). The hyperbolic plane is a formula_9-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.
Another family of examples of cocompact Fuchsian groups is given by triangle groups: all but finitely many are hyperbolic.
Negative curvature.
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 of a form of signature formula_50 are hyperbolic.
A further generalisation is given by groups admitting a geometric action on a CAT(k) space. There exist examples which are not commensurable to any of the previous constructions (for instance groups acting geometrically on hyperbolic buildings).
Small cancellation groups.
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.
Random groups.
In some sense, "most" finitely presented groups with large defining relations are hyperbolic. For a quantitative statement of what this means see Random group.
Generalisations.
Relatively hyperbolic groups.
Relatively hyperbolic groups are a class generalising hyperbolic groups. "Very" roughly formula_1 is hyperbolic relative to a collection formula_62 of subgroups if it admits a ("not necessarily cocompact") properly discontinuous action on a proper hyperbolic space formula_2 which is "nice" on the boundary of formula_2 and such that the stabilisers in formula_1 of points on the boundary are subgroups in formula_62. This is interesting when both formula_2 and the action of formula_1 on formula_2 are not elementary (in particular formula_2 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. Non-examples are lattices in higher-rank Lie groups and mapping class groups.
Acylindrically hyperbolic groups.
An even more general notion is that of an acylindically hyperbolic group. Acylindricity of an action of a group formula_1 on a metric space formula_2 is a weakening of proper discontinuity of the action.
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.
CAT(0) groups.
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).