Abstract
The group of direct isometries of the hyperbolic space \({\Bbb H}^n \mbox{is} G=\mbox{SO}_0(n,1).\) This isometric action admits many differentiable compactifications into an action on the closed n-dimensional ball. We prove that all such compactifications are topologically conjugate but not necessarily differentiably conjugate. We give the classifications of real analytic and smooth compactifications.
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