Maps between Relatively Hyperbolic Boundaries
Maps between Relatively Hyperbolic Boundaries
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 …