Gromov hyperbolicity of the $j_G$ and ${\tilde \jmath }_G$ metrics

Abstract

In this note it is shown that the ${\tilde \jmath }_G$ metric is always Gromov hyperbolic, but that the $j_G$ metric is Gromov hyperbolic if and only if $G$ has exactly one boundary point. As a corollary we get a new proof for the fact that the quasihyperbolic metric is Gromov hyperbolic in uniform domains.