## Locally SU6(C) Graphs

### (Joint work with Ralf Gramlich)

Lecturer: Kristina Altmann, Technische Universitaet Darmstadt

 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 Un over C endowed with a non-degenerate hermitian form and define the graph G(Un) on the lines of Un 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(Un) then Γ is isomorphic toG(Un+2) or to the graph on the fundamental SU2's of 2E6(C). As a corollary this leads to a characterisation of the groups SUn+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