|Algebraic structure → Group theory|
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 ) 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.
Let be a finitely generated group, and be its Cayley graph with respect to some finite set of generators. The 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. The group is then said to be hyperbolic if is a hyperbolic space in the sense of Gromov. Shortly, this means that there exists a such that any geodesic triangle in is -thin, as illustrated in the figure on the right (the space is then said to be -hyperbolic).
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 being hyperbolic without referring to a generating set. On the other hand, a space which is quasi-isometric to a -hyperbolic space is itself -hyperbolic for some but the latter depends on both the original and on the quasi-isometry, thus it does not make sense to speak of being -hyperbolic.
The Švarc–Milnor lemma states that if a group acts properly discontinuously and with compact quotient (such an action is often called geometric) on a proper length space , then it is finitely generated, and any Cayley graph for is quasi-isometric to . Thus a group is (finitely generated and) hyperbolic if and only if it has a geometric action on a proper hyperbolic space.
If is a subgroup with finite index (i.e., the set is finite), then the inclusion induces a quasi-isometry on the vertices of any locally finite Cayley graph of into any locally finite Cayley graph of . Thus is hyperbolic if and only if itself is. More generally, if two groups are commensurable, then one is hyperbolic if and only if the other is.
The simplest examples of hyperbolic groups are finite groups (whose Cayley graphs are of finite diameter, hence -hyperbolic with equal to this diameter).
Another simple example is given by the infinite cyclic group : the Cayley graph of with respect to the generating set is a line, so all triangles are line segments and the graph is -hyperbolic. It follows that any group which is virtually cyclic (contains a copy of 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).
Let be a finite set and be the free group with generating set . Then the Cayley graph of with respect to is a locally finite tree and hence a 0-hyperbolic space. Thus is a hyperbolic group.
More generally we see that any group which acts properly discontinuously on a locally finite tree (in this context this means exactly that the stabilizers in of the vertices are finite) is hyperbolic. Indeed, this follows from the fact that 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 : 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 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 is not quasi-isometric to the hyperbolic plane).
Main article: Fuchsian group
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 ). The hyperbolic plane is a -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.
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 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).
Main article: Small cancellation theory
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.
Main article: Random group
In some sense, "most" finitely presented groups with large defining relations are hyperbolic. For a quantitative statement of what this means see Random group.
Main article: Relatively hyperbolic group
Relatively hyperbolic groups are a class generalising hyperbolic groups. Very roughly is hyperbolic relative to a collection of subgroups if it admits a (not necessarily cocompact) properly discontinuous action on a proper hyperbolic space which is "nice" on the boundary of and such that the stabilisers in of points on the boundary are subgroups in . This is interesting when both and the action of on are not elementary (in particular 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.
An even more general notion is that of an acylindically hyperbolic group. Acylindricity of an action of a group on a metric space 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.
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).