The Gromov boundary of a hyperbolic group is the space of asymptotic classes of rays in its Cayley
graph with the visual topology. Each ray is a distance-preserving
path from 
into the graph starting
at a fixed vertex. Two such rays determine the same
point of the Gromov boundary when their images remain within
a bounded distance of each other. The topology records
agreement to increasing distances from the basepoint.
Changing the finite generating set
produces a homeomorphic boundary.
For a free group on at least two generators, the boundary is a Cantor set. The boundary of the infinite cyclic group consists of two points. The Cannon conjecture instead concerns a boundary homeomorphic to a two-dimensional sphere.
