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 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 ${\displaystyle G}$ be a finitely generated group, and ${\displaystyle X}$ be its Cayley graph with respect to some finite set ${\displaystyle S}$ of generators. The set ${\displaystyle X}$ 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 ${\displaystyle G}$ is then said to be hyperbolic if ${\displaystyle X}$ is a hyperbolic space in the sense of Gromov. Shortly, this means that there exists a ${\displaystyle \delta >0}$ such that any geodesic triangle in ${\displaystyle X}$ is ${\displaystyle \delta }$-thin, as illustrated in the figure on the right (the space is then said to be ${\displaystyle \delta }$-hyperbolic).

A priori this definition depends on the choice of a finite generating set ${\displaystyle S}$. That this is not the case follows from the two following facts:

Thus we can legitimately speak of a finitely generated group ${\displaystyle G}$ being hyperbolic without referring to a generating set. On the other hand, a space which is quasi-isometric to a ${\displaystyle \delta }$-hyperbolic space is itself ${\displaystyle \delta '}$-hyperbolic for some ${\displaystyle \delta '>0}$ but the latter depends on both the original ${\displaystyle \delta }$ and on the quasi-isometry, thus it does not make sense to speak of ${\displaystyle G}$ being ${\displaystyle \delta }$-hyperbolic.

The Švarc–Milnor lemma states that if a group ${\displaystyle G}$ acts properly discontinuously and with compact quotient (such an action is often called geometric) on a proper length space ${\displaystyle Y}$, then it is finitely generated, and any Cayley graph for ${\displaystyle G}$ is quasi-isometric to ${\displaystyle Y}$. Thus a group is (finitely generated and) hyperbolic if and only if it has a geometric action on a proper hyperbolic space.

If ${\displaystyle G'\subset G}$ is a subgroup with finite index (i.e., the set ${\displaystyle G/G'}$ is finite), then the inclusion induces a quasi-isometry on the vertices of any locally finite Cayley graph of ${\displaystyle G'}$ into any locally finite Cayley graph of ${\displaystyle G}$. Thus ${\displaystyle G'}$ is hyperbolic if and only if ${\displaystyle G}$ 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 ${\displaystyle \delta }$-hyperbolic with ${\displaystyle \delta }$ equal to this diameter).

Another simple example is given by the infinite cyclic group ${\displaystyle \mathbb {Z} }$: the Cayley graph of ${\displaystyle \mathbb {Z} }$ with respect to the generating set ${\displaystyle \{\pm 1\}}$ is a line, so all triangles are line segments and the graph is ${\displaystyle 0}$-hyperbolic. It follows that any group which is virtually cyclic (contains a copy of ${\displaystyle \mathbb {Z} }$ of finite index) is also hyperbolic, for example the infinite dihedral group.