Hyperbolic group

In group theory, a hyperbolic group is a special kind of group that follows certain rules related to distances between its elements. This idea comes from a branch of mathematics called geometric group theory. Think of it as a way to measure how far apart different elements of a group are from each other.

The concept of a hyperbolic group was introduced by a mathematician named Mikhail Gromov in 1987. He was inspired by ideas from hyperbolic geometry, the study of shapes and spaces that bend in unusual ways. Gromov also drew from the study of shapes in three dimensions and from combinatorial group theory, which looks at groups using rules and patterns.

Many other mathematicians have contributed to the development of this theory. Their work helps us understand more about how groups behave and how they can be used in different areas of mathematics.

Definition

A hyperbolic group is a special kind of group in mathematics. Groups are collections of things that can be combined in a specific way. To understand hyperbolic groups, we look at something called a Cayley graph. This graph shows how the elements of the group connect to each other.

We say a group is hyperbolic if its Cayley graph follows certain rules that make it similar to space in hyperbolic geometry. This means that any triangle formed by the shortest paths in the graph stays close to being thin. This idea was introduced by a mathematician named Mikhail Gromov in 1987.

The definition of a hyperbolic group does not depend on how we choose the generators, which are the basic elements we use to build the group. This is because different choices of generators lead to graphs that look almost the same to mathematicians, called quasi-isometric.

Examples

Elementary hyperbolic groups

The simplest examples of hyperbolic groups are finite groups. Another simple example is the infinite cyclic group Z: its Cayley graph with respect to the generating set { ± 1 } is a line, so all triangles are line segments and the graph is 0-hyperbolic. Any group which is virtually cyclic (contains a copy of Z of finite index) is also hyperbolic, for example the infinite dihedral group.

Free groups and groups acting on trees

Let S = { a₁, … , aₙ } be a finite set and F be the free group with generating set S. Then the Cayley graph of F with respect to S is a locally finite tree and hence a 0-hyperbolic space. Thus F is a hyperbolic group.

More generally, any group G which acts properly discontinuously on a locally finite tree (in this context this means exactly that the stabilizers in G of the vertices are finite) is hyperbolic. 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 G = SL₂(Z): 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.

Fuchsian groups

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 SL₂(R)). The hyperbolic plane is a δ-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 preserving a form of signature ( n , 1 ) are hyperbolic.

A further generalisation is given by groups admitting a geometric action on a CAT(k) space, when k is any negative number. There exist examples which are not commensurable to any of the previous constructions (for instance groups acting geometrically on hyperbolic buildings).

Small cancellation groups

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.

Random groups

Main article: Random group

In some sense, "most" finitely presented groups with large defining relations are hyperbolic.

Non-examples

The simplest example of a group which is not hyperbolic is the free rank 2 abelian group Z². Indeed, it is quasi-isometric to the Euclidean plane which is easily seen not to be hyperbolic.

More generally, any group which contains Z² as a subgroup is not hyperbolic. In particular, lattices in higher rank semisimple Lie groups and the fundamental groups π₁(S³ ∖ K) of nontrivial knot complements fall into this category and therefore are not hyperbolic. This is also the case for mapping class groups of closed hyperbolic surfaces.

The Baumslag–Solitar groups B(m,n) and any group that contains a subgroup isomorphic to some B(m,n) fail to be hyperbolic.

A non-uniform lattice in a rank 1 simple Lie group is hyperbolic if and only if the group is isogenous to SL₂(R) (equivalently the associated symmetric space is the hyperbolic plane). An example of this is given by hyperbolic knot groups. Another is the Bianchi groups, for example SL₂(−1).

Properties

Hyperbolic groups have special rules and patterns that make them interesting to mathematicians. One important rule is called the Tits alternative, which says these groups either behave in a simple way or contain a part that looks like a non-abelian free group.

These groups also have geometric properties. For example, when they are large and not almost cyclic, they grow at an exponential rate. They also follow a special rule called a linear isoperimetric inequality.

In terms of algebra, hyperbolic groups can always be described with a finite list of rules. They also have solutions to certain problems, like the word problem, and can be studied using special structures and rules.

Generalisations

Main article: Relatively hyperbolic group

Relatively hyperbolic groups are a broader class that builds on hyperbolic groups. They involve a group acting in a special way on a certain kind of space, which helps to study more complex mathematical structures.

Another idea is that of an acylindrically hyperbolic group. This is an even broader concept, where the group acts on a space in a way that is less strict but still follows certain rules. This includes groups that act on special structures called curve complexes.

There are also groups called CAT(0) groups, which act on spaces with certain curvature properties. This includes groups that act on Euclidean space in a regular pattern. It is still an open question whether every hyperbolic group is also a CAT(0) group.