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 to**G**(U_{n+2}) or to the graph on the fundamental SU_{2}'s of ^{2}E_{6}(**C**).
As a corollary this leads to a characterisation of the groups SU_{n+2}(**C**), *n ≥ 6*, and ^{2}E_{6}(**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 |