9 - Locally projective graphs

Summary

In this chapter we study locally projective graphs. Let Γ be a graph and G be a vertex-transitive automorphism group of Γ. Then Γ is said to be a locally projective graph with respect to G if for every x ∈ Γ the subconstituent G(x)^Γ(x) is a projective linear group in its natural permutation representation. Incidence graphs of certain truncations of classical geometries are locally projective graphs with respect to their full automorphism groups. These examples can be characterized in the class of all locally projective graphs by the property that their girth is a small even number. We present a proof of this characterization based on the classification of Tits geometries and observe how a class of sporadic Petersen geometries naturally appear in this context via locally projective graphs of girth 5. In Section 9.1 we review some basic results on 2-arc-transitive actions of groups on graphs. In Section 9.2 we discuss examples of locally projective graphs coming from classical geometries. Locally projective lines and their characterizations are discussed in Section 9.3. In Section 9.4 we analyse the possibilities for the action of the vertex stabilizer G(x) on the set of vertices at distance 2 from x. These possibilities determine the main types of locally projective graphs. In a locally projective graph there are virtual projective space structures defined on neighbourhoods of vertices. These virtual structures lead to the notion of geometrical subgraphs introduced in Section 9.5.