Local recognition of non-incident point-hyperplane graphs

Let P be a projective space. By H(P) we denote the graph whose vertices are the non-incident point-hyperplane pairs of P, two vertices (p,H) and (q, I) being adjacent if and only if p ∈ I and q ∈ H. In this paper we give a characterization of the graph H(P) (as well as of some related graphs) by its local structure. We apply this result by two characterizations of groups G with PSLn(F) ≤ G ≤ PGLn(F), by properties of centralizers of some (generalized) reflections. Here F is the (skew) field of coordinates of P