A hyperbolic group is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry.
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.
Let $G$ be a finitely generated group, and $X$ be its Cayley graph with respect to some finite set $S$ of generators. The set $X$ 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 $G$ is then said to be hyperbolic if $X$ is a hyperbolic space in the sense of Gromov. Shortly, this means that there exists a $\delta >0$ such that any geodesic triangle in $X$ is $\delta$-thin (the space is then said to be $\delta$-hyperbolic).
A priori this definition depends on the choice of a finite generating set $S$. That this is not the case follows from the two following facts:
- the Cayley graphs corresponding to two finite generating sets are always quasi-isometric one to the other;
- any geodesic space which is quasi-isometric to a geodesic Gromov-hyperbolic space is itself Gromov-hyperbolic.
Thus we can legitimately speak of a finitely generated group $G$ being hyperbolic without referring to a generating set. On the other hand, a space which is quasi-isometric to a $\delta$-hyperbolic space is itself $\delta '$-hyperbolic for some $\delta '>0$ but the latter depends on both the original $\delta$ and on the quasi-isometry, thus it does not make sense to speak of $G$ being $\delta$-hyperbolic.
Source: Wikipedia
