Jørgensen’s inequality for metric spaces with application to the octonions

Jørgensen’s inequality gives a necessary condition for the discreteness of a non-elementary group of isometries of hyperbolic 3-space. The main idea of the proof may be generalised widely but the statement is quite specialised. Here we give a scheme for restating Jørgensen’s inequality for Möbius transformations of a metric space. This unifies many previously published versions of Jørgensen’s inequality. We then show how this scheme may be applied by giving a version of Jørgensen’s inequality for the octonionic hyperbolic plane.