On the uniqueness of the \(Co_ 1\) 2-local geometry (Q1103192)

In this remarkable paper the author shows the uniqueness of a certain geometry for \(Co_ 1\) discovered by \textit{M. A. Ronan} and \textit{S. D. Smith} [Proc. Symp. Pure Math. 37, 283-289 (1980; Zbl 0478.20015)]. This is a geometry consisting of points, lines, planes and hyperplanes such that the residue of a point is a \(M_{24}\)- geometry, also discovered by M. Ronan and S. Smith, and the residue of a hyperplane consists of the points, lines and one class of maximal singular subspaces of the \(0\) \(+_ 8(2)\)- building. The author shows that a geometry \(\Gamma\) having this kind of residue structure and possessing a flag transitive automorphism group G, is uniquely determined and \(G\cong Co_ 1\). The proof of this result is divided into two parts. First the stabilizers of a point and a hyperplane are determined. Then it is shown that G admits a rank 6 representation on the points. Using this information it is possible to show that the collinearity graph of \(\Gamma\) is uniquely determined.