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This article provides an almost self-contained, purely combinatorial local recognition of the graph on the non-intersecting line-hyperline pairs of the projective space PnðFÞ for nd 8 and F a division ring with the exception of the case n 1⁄4 8 and F 1⁄4 F2. Consequences of that result are a characterization of the hyperbolic root group geometry of SLnþ1ðFÞ, F a division ring, and a local recognition of certain groups containing a central extension of PSLnþ1ðFÞ, F a field, using centralizers of p-elements.

Line-hyperline pairs of projective spaces and fundamental subgroups of linear groups