A hyperbolic group, also called a word-hyperbolic group, is a finitely generated group whose Cayley
graph, with edges of length
1, has uniformly thin triangles formed by distance-minimizing
paths. There is a constant 
such
that every point of each side lies within distance 
of the union
of the other two sides. The property is independent of the finite
generating set. Free groups
provide examples because their standard Cayley graphs
are trees.
The geometry at infinity is described by the Gromov boundary. The Cannon conjecture concerns groups for which that boundary is a two-dimensional sphere.
