A Graph with a Locally Projective Vertex-Transitive Group of Automorphisms Aut (Fi22) Which Has a Nontrivial Stabilizer of a Ball of Radius 2

Earlier, to confirm that one of the possibilities for the structure of vertex stabilizers of graphs with projective suborbits is realizable, we announced the existence of a connected graph Gamma admitting a group of automorphisms G which is isomorphic to Aut(Fi(22)) and has the following properties. First, the group G acts transitively on the set of vertices of Gamma, but intransitively on the set of 3-arcs of Gamma. Second, the stabilizer in G of a vertex of Gamma induces on the neighborhood of this vertex a group PSL3(3) in its natural doubly transitive action. Third, the pointwise stabilizer in Gof a ball of radius 2 in Gamma is nontrivial. In this paper, we construct such a graph Gamma with G = Aut(Gamma).