A central problem in synthetic geometry is the characterisation of graphs and geometries. The local recognition of locally homogeneous graphs forms one category of such characterisations: Choose a graph Δ, and try to identify all connected graphs which are locally Δ. We focus on the n-dimensional vector space U_n over C endowed with a non-degenerate hermitian form and define the graph G(U_n) on the lines of U_n where two lines l and m are adjacent if and only if l is perpendicular to m with respect to the hermitian form.

Let n ≥ 6 and Γ be a connected graph which is locally G(U_n) then Γ is isomorphic toG(U_n+2) or to the graph on the fundamental SU₂'s of ²E₆(C). As a corollary this leads to a characterisation of the groups SU_n+2(C), n ≥ 6, and ²E₆(C), see Theorem 27.1 of [1].

References:

[1] D.Gorenstein, R Lyons, R.Solomon, The Classification of the Finite Simple Groups, American Mathematical Society, 1994
[2] R.Gramlich, On Graphs, Geometries and Groups of Lie Type, Eindhoven University Press, 2002