Hyperbolic group

In group theory, a hyperbolic group is a special kind of group. It has rules about the distances between its elements. This idea comes from a branch of mathematics called geometric group theory.

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 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 sets of things that can be combined in a certain way. To understand hyperbolic groups, we look at something called a Cayley graph. This graph shows how the elements of the group connect.

We say a group is hyperbolic if its Cayley graph follows special rules that make it like space in hyperbolic geometry. This means that any triangle made from 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 which basic elements, called generators, we use to build the group. This is because different choices of generators create graphs that look almost the same to mathematicians, called quasi-isometric.

Examples

Elementary hyperbolic groups

The simplest hyperbolic groups are finite groups. Another example is the infinite cyclic group Z. Its Cayley graph looks like a straight line, so it is 0-hyperbolic. Any group that is virtually cyclic (contains a copy of Z with finite index) is also hyperbolic, such as 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. The Cayley graph of F with respect to S is a tree, which is 0-hyperbolic. So F is a hyperbolic group.

More generally, any group G that acts properly on a tree is hyperbolic. These groups are virtually free and are also hyperbolic.

An example is the modular group G = SL₂(Z). It acts on a tree and has a free subgroup.

Fuchsian groups

Main article: Fuchsian group

A Fuchsian group is a group that acts on the hyperbolic plane. The hyperbolic plane is δ-hyperbolic, so cocompact Fuchsian groups are hyperbolic.

Examples include the fundamental groups of closed surfaces with negative Euler characteristic. These surfaces are pieces of the hyperbolic plane.

Another example is triangle groups. Most of these are hyperbolic.

Negative curvature

The fundamental groups of certain Riemannian manifolds with negative sectional curvature are hyperbolic. Examples include lattices in the orthogonal or unitary group.

Groups that act on a CAT(k) space with k negative are also hyperbolic.

Small cancellation groups

Main article: Small cancellation theory

Groups with presentations that meet small cancellation rules are hyperbolic.

Random groups

Main article: Random group

Most finitely presented groups with many defining relations are hyperbolic.

Non-examples

The free rank 2 abelian group Z² is not hyperbolic. It is similar to the Euclidean plane, which is not hyperbolic.

Any group that includes Z² as a subgroup is not hyperbolic. This includes lattices in higher rank semisimple Lie groups and fundamental groups of knot complements.

The Baumslag–Solitar groups B(m,n) and groups containing them are not hyperbolic.

A non-uniform lattice in a rank 1 simple Lie group is hyperbolic only if it is isogenous to SL₂(R). Examples include hyperbolic knot groups and the Bianchi groups, like SL₂(−1).

Properties

Hyperbolic groups have special rules that make them interesting to mathematicians. One important rule is called the Tits alternative. It 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 special shapes. When they are large and not almost cyclic, they grow very fast. They also follow a special rule called a linear isoperimetric inequality.

In algebra, hyperbolic groups can always be described with a short list of rules. They also have answers 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 bigger group family that builds on hyperbolic groups. They act 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 bigger concept, where the group acts on a space in a less strict but still rule-following way. 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 unknown if every hyperbolic group is also a CAT(0) group.