Flag-transitive \(c\)-extensions of the \(F_4(2)\)-building -- the amalgams
type: Item
instance of: scholarly article
title: Flag-transitive \(c\)-extensions of the \(F_4(2)\)-building -- the amalgams
zbMATH Open document ID: 1111.51009
author: Corinna Wiedorn
published in: European Journal of Combinatorics
publication date: 2007-02-14T00:00:00Z
review text: 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 e…
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20E42
zbMATH DE Number: 5126367
zbMATH Keywords: building
zbMATH Keywords: geometry
zbMATH Keywords: diagram
zbMATH Keywords: flag-transitive automorphism group
zbMATH Keywords: amalgam
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
full work available at URL: j.ejc.2005.06.008
OpenAlex ID: W2092924810
cites work: Affine polar spaces
cites work: Q3684278
cites work: Automorphisms of trivalent graphs
cites work: A characterization of the groups \(F_ 4(2^ n)\)
cites work: \(c\)-extensions of the \(F_{4}\)(2)-building
cites work: Extended \(F_4\)-buildings and the Baby Monster
cites work: A diameter bound for extensions of the \(F_{4}\)(2)-building
cites work: Geometry of sporadic groups. II: Representations and amalgams
cites work: Amalgams Determined By Locally Projective Actions
cites work: More on geometries of the Fischer group \(Fi_{22}\).
cites work: Q4114031
DOI: 10.1016/J.EJC.2005.06.008
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description: scientific article; zbMATH DE number 5126367