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

Jorgensen’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 Jorgensen’s inequality for Mobius transformations of a metric space. This unifies many previously published versions of Jorgensen’s inequality. We then show how this scheme may be applied by giving a version of Jorgensen’s inequality for the octonionic hyperbolic plane.