Two sporadic geometries related to the Hoffman-Singleton graph (Q908512)

The main result of the paper is as follows: Up to isomorphism there exists a unique residually connected Tits geometry \(\Gamma^{(i)}\) belonging to \(\Delta^{(i)}\) \((i=1,2)\), such that every rank 3 residue belonging to the diagram \(C_ 3\) is the sporadic \(A_ 7\)-geometry. Although the real contribution is the uniqueness, the paper also contains constructions in order to give motivation for the uniqueness proof.