  In [BBM21], Belk, Bleak and Matucci proved that hyperbolic groups can be seen
as subgroups of the rational group. In order to do so, they associated a tree
of atoms to each hyperbolic group. Not so many connections between this tree
and the literature on hyperbolic groups were known. In this paper, we prove an
atom-version of the fellow traveler property and exponential divergence,
together with other similar results. These leads to several consequences: a
bound from above of the topological dimension of the Gromov boundary, the
definition of an augmented tree which is quasi-isometric to the Cayley graph
and a synchronous recognizer which described the equivalence relation given by
the quotient map defined from the end of the tree onto the Gromov boundary.
