  Let $G_1,G_2$ be two word hyperbolic groups and $f:\partial G_1\to \partial
G_2$ be a homeomorphism between their Gromov boundaries. F. Paulin proved that
if $f$ is a quasi-M\"obius equivalence then there exists a quasi-isometry $\Phi
f: G_1\to G_2$ such that $\Phi f$ induces the homeomorphism $f$ between the
Gromov boundaries of $G_1$ and $G_2$. The proof given by Paulin was written in
French, keeping an eye for broad readers, we will first reproduce Paulin's work
and then extend Paulin's results to relatively hyperbolic groups. Given two
relatively hyperbolic groups, we will show that a cusp preserving
quasi-isometry induces a homeomorphism between their relatively hyperbolic
boundaries preserving parabolic endpoints and the induced homeomorphism
linearly distorts the exit points of bi-infinite geodesics to combinatorial
horoballs. Conversely, we will then show that a homeomorphism between
relatively hyperbolic boundaries preserving parabolic endpoints and distorting
the exit points of bi-infinite geodesics to combinatorial horoballs linearly
will induce a cusp preserving quasi-isometry between the relatively hyperbolic
groups.
