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 PSL _n ($\Bbb F$)≤G≤PGL _n ($\Bbb F$), by properties of centralizers of some (generalized) reflections. Here $\Bbb F$ is the (skew) field of coordinates of ℙ.