Tits endomorphisms and buildings of type F 4
[Endomorphismes de Tits et immeubles de type F 4 ]
Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2349-2421.

L’immeuble de points fixes d’une polarité d’un quadrangle de Moufang de type F 4 est un ensemble de Moufang. Il en va de même pour l’immeuble de points fixes d’un automorphisme semi-linéaire d’ordre 2 d’un octogone de Moufang qui stabilise au moins deux cloisons d’un type mais aucun de l’autre. Nous montrons que ces deux classes d’ensembles de Moufang sont en fait identiques, que chaque membre de cette classe peut être construit comme l’immeuble de points fixes d’un groupe d’ordre 4 agissant sur un immeuble de type F 4 , et que pour chacun de ces ensembles de Moufang, le groupe engendré par tous les sous-groupes radiciels est un groupe simple.

The fixed point building of a polarity of a Moufang quadrangle of type F 4 is a Moufang set, as is the fixed point building of a semi-linear automorphism of order 2 of a Moufang octagon that stabilizes at least two panels of one type but none of the other. We show that these two classes of Moufang sets are, in fact, the same, that each member of this class can be constructed as the fixed point building of a group of order 4 acting on a building of type F 4 and that the group generated by all the root groups of any one of these Moufang sets is simple.

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DOI : https://doi.org/10.5802/aif.3138
Classification : 20E42,  51E12,  51E24
Mots clés : immeubles, descent, polarité, ensemble de Moufang, quadrangle de Moufang, octogone de Moufang
@article{AIF_2017__67_6_2349_0,
     author = {De Medts, Tom and Segev, Yoav and Weiss, Richard M.},
     title = {Tits endomorphisms and buildings of type~$F_4$},
     journal = {Annales de l'Institut Fourier},
     pages = {2349--2421},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {6},
     year = {2017},
     doi = {10.5802/aif.3138},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3138/}
}
De Medts, Tom; Segev, Yoav; Weiss, Richard M. Tits endomorphisms and buildings of type $F_4$. Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2349-2421. doi : 10.5802/aif.3138. http://archive.numdam.org/articles/10.5802/aif.3138/

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