On the \(Fi_{22}\)-minimal parabolic geometry (Q1919268)

The \(Fi_{22}\)-minimal parabolic geometry is a residually connected flag transitive string geometry with flag transitive group \(G\) such that the residues which are not digons are isomorphic to either the geometry of duads and triduads on the Steiner system \(S(22, 3,6)\) (and \(G\) induces the sporadic simple group \(M_{22}\) on each such residue; also, in this geometry every duad is incident with 15 triduads and every triduad contains 3 duads), or to the geometry obtained from the unique generalized quadrangle of order (4,2) by taking three isomorphic copies of it and identifying lines which correspond under the isomorphisms considered, but not the points (hence this geometry has 135 points and 27 lines; each line is incident with 15 points and each point with 3 lines). The residues which are generalized digons contain 3 elements of each type. The main result of the paper under review is that, when a (not necessarily finite) geometry \(\Gamma\) satisfies the above assumptions, and the stabilizer of a maximal flag is finite, then \(G\) is isomorphic to the sporadic group \(Fi_{22}\) and \(\Gamma\) is the minimal parabolic geometry for \(Fi_{22}\). Much geometry is used in the proof, and consequently, a lot of geometric properties follow from this characterization.