It is well known that a subgroup of $PSL(2,\mathbb{R})$ is discrete iff it acts properly discontinuously on the hyperbolic plane $\mathbb{H}$.
I wounder - does the same hold for a discrete group of isometries of $\mathbb {H}^n$ for $n>2$?
Also, a reference regarding hyperbolic isometries of high dimension would be of help, even not only about this specific question.