Flag-transitive \(c\)-extensions of the \(F_4(2)\)-building -- the amalgams
Purely combinatorial characterizations of \(c\)-extensions of buildings of type \(F_4(t)\), \(t=1,2,4\) were obtained under certain additional assumptions by \textit{A. Ivanov, D. Pasechnik} and \textit{S. Shpectorov} [Invent. Math. 144, 399--433 (2001; Zbl 0990.20007)] for \(t=4\) and \textit{A. Ivanov} and the author [J. Algebr. Comb. 16, 111--150 (2002; Zbl 1059.51015)] for \(t=1\). In the former case there is a unique geometry; its automorphism group is flag-transitive and isomorphic to the Baby Monster. In the latter case two geometries exist with the sporadic simple group \(Fi_{22}\) and its non-split extension \(3\cdot Fi_{22}\) as flag-transitive automorphism groups. In the paper under review the author contributes to the remaining case \(t=2\) where it is assumed that (a) any two elements of type 0 (whose residues are buildings of type \(F_4(2)\)) are incident to at most one common element of type 1; (ab) if two elements of type 0 are incident to a common element \(x\) of type 1 and to a common element \(y\) of type 4, then \(x\) and \(y\) are incident; and (b) any three elements of type 0 are pairwise indcident to a common element of type 1 if and only if all three are incident to a common element of type 4. Even under these addtinonal assumptions the flag-transitive geometries have not been completely classifed yet and there are only four such geometries known with automorphism groups \(^2E_6(2):2\), \(3\cdot{^2E_6(2):2}\), \(E_6(2):2\), and \(2^{26}:F_4(2)\). The author considers a geometry \({\mathcal G}\) that is a \(c\)-extension of a building of type \(F_4(2)\) satisfying conditions (a), (ab) and (b) and a flag-transitive group \(G\) of automorphisms of \({\mathcal G}\). In fact, it is shown that (a) is a consequence of flag-transitivity. She investigates amalgams \({\mathcal A}=\{G_j\mid 0\leq j\leq 4\}\) of maximal parabolics associated with the action of \(G\) on \({\mathcal G}\) and shows that \(G_0\), the stabilizer of an element of type 0, is either \(F_4(2)\) or \(F_4(2)\times 2\). Considering subamalgams \(\{G_0,G_1\}\) the author gives a list of possibilities for \(G_1\). As a consequence she obtains that besides \(G\cong 2^{26}:F_4(2)\) there are at most eight possibilities for the isomorphism type of \({\mathcal A}\).
scientific article; zbMATH DE number 5126367