An Essay on the Ree Octagons
Michael Joswig
and H. Van Maldeghem
DOI: 10.1023/A:1022481514132
Abstract
We coordinatize the Moufang generalized octagons arising from the Ree groups of type 2 F 4. In this way, we obtain a very concrete and explicit description of these octagons. We use this to prove some results on suboctagons, generalized homologies, Suzuki-Tits ovoids and groups of projectivities of the Ree octagons. All our results hold for arbitrary Ree octagons, finite or not.
Pages: 145–164
Keywords: Tits building; generalized octagon; group of projectivities; coordinatization
Full Text: PDF
References
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19. H. Van Maldeghem "Generalized polygons with valuation," Arch. Math. 53 (1983), 513-520.
20. H. Van Maldeghem, "A configurational characterization of the Moufang generalized polygons," European J. Combin. 11 (1990), 362-372.
21. H. Van Maldeghem, "Common characterizations of the finite Moufang polygons," Proc. Third Isle of Thorn Conference on Finite Geometries and Designs, Adv. Finite Geom. and Designs (1991), 391 -400.
22. H. Van Maldeghem, "The geometry of traces in Ree octagons," to appear in the proceedings of Groups of Lie type and their geometries, Como, 1993.
23. H. Van Maldeghem and R. Weiss, "On finite Moufang polygons," Israel J. Math. 79 (1992), 321-330.
24. R. Weiss, "The nonexistence of certain Moufang polygons," Inv. Math. 51(1979), 261-266.
2. W. Feit and G. Higman, "The non-existence of certain generalized polygons," J. Algebra 1 (1964), 114-131.
3. G. Hanssens and H. Van Maldeghem, "Coordinatization of generalized quadrangles," Ann. Discr. Math. 37 (1988), 195-208.
4. G. Hanssens and H. Van Maldeghem, "Algebraic properties of quadratic quaternary rings," Geom. Ded. 30 (1989), 43-67.
5. N. Knarr, "Projectivities of generalized polygons," Ar.i Comb. 25B (1988), 265-275.
6. N. Knarr, "The nonexistence of certain topological polygons," Forum Math. 2 (1990), 603-612.
7. S. Lowe, "Uber die Konstruktion von verallgemeinerten Vierecken mit einem regularen Punkt," Habilitationsschrift, Braunschweig (1992).
8. S.E Payne and J.R Thas, Finite generalized quadrangles, Pitman, 1984.
9. M. Ronan, Lectures on buildings, Academic Press, Persp. Math. 7, 1989.
10. J. Sarli, "The geometry of root subgroups in Ree groups of type 2Fn" Geom. Ded. 26 (1988), 1-28.
11. J.A Thas, "A restriction on the parameters of a suboctagon," J. Combin. Th. A 27(1979), 385-387. 12.
1. Tits, "Sur la trialitd et certains groupes qui s'en deduisent," Publ. Math. IHES2 (1959), 14-60.
13. J. Tits, "Les groupes simples de Suzuki et de Ree," Sent. Bourbaki exp. 210 (1960).
14. J. Tits, "Ovoi'des et groupes de Suzuki," Arch. Math. 13 (1962), 187-198.
15. J. Tits, "Buildings of spherical type and finite B/V-pairs," Springer, Lecture Notes Math. 386, 1974.
16. J. Tits, "Quadrangles de Moufang. I," preprint (1976).
17. J. Tits, "Non-existence de certains polygones generalisee I/II," Inv. Math. 36 (1976), 275-284; Inv. Math. 51 (1979), 267-269.
18. J. Tits, "Moufang octagons and the Ree groups of type 2F4," Am. J. Math. 105 (1983), 539-594.
19. H. Van Maldeghem "Generalized polygons with valuation," Arch. Math. 53 (1983), 513-520.
20. H. Van Maldeghem, "A configurational characterization of the Moufang generalized polygons," European J. Combin. 11 (1990), 362-372.
21. H. Van Maldeghem, "Common characterizations of the finite Moufang polygons," Proc. Third Isle of Thorn Conference on Finite Geometries and Designs, Adv. Finite Geom. and Designs (1991), 391 -400.
22. H. Van Maldeghem, "The geometry of traces in Ree octagons," to appear in the proceedings of Groups of Lie type and their geometries, Como, 1993.
23. H. Van Maldeghem and R. Weiss, "On finite Moufang polygons," Israel J. Math. 79 (1992), 321-330.
24. R. Weiss, "The nonexistence of certain Moufang polygons," Inv. Math. 51(1979), 261-266.