More on Geometries of the Fischer Group Fi 22
A. A. Ivanov1
and C. Wiedorn2
1Department of Mathematics Imperial College 180 Queens Gate London o[SW7 2BZ
2Department of Mathematics University of Birmingham Edgbaston Birmingham B15 2TT
2Department of Mathematics University of Birmingham Edgbaston Birmingham B15 2TT
DOI: 10.1023/A:1021620911639
Abstract
We give a new, purely combinatorial characterization of geometries e ε with diagram identifying each under some E( Fi 22 ) \mathcal{E}(Fi_{22} ) and E(3 \cdot Fi 22 ) \mathcal{E}(3 \cdot Fi_{22} ) related to the Fischer 3-transposition group Fi 22 and its non-split central extension 3 ; Fi 22, respectively. As a by-product we improve the known characterization of the c-extended dual polar spaces for Fi 22 and 3 ; Fi 22 and of the truncation of the c-extended 6-dimensional unitary polar space.
Pages: 111–150
Keywords: fischer group; diagram geometry; extended building
Full Text: PDF
References
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2. H. Cuypers, “Finite locally generalized quadrangles with affine planes,” Europ. J. Comb. 13 (1992), 439-453.
3. J. Hall and S.V. Shpectorov, “Rank 3 P-geometries,” Geom. Dedicata. 82 (2000), 139-169.
4. A.A. Ivanov, S.A. Linton, K. Lux, J. Saxl, and L.H. Soicher, “Distance-transitive representations of the sporadic groups,” Comm. Algebra 23(9) (1995), 3379-3427.
5. A.A. Ivanov, D.V. Pasechnik, and S.V. Shpectorov, “Extended F4-buildings and the Baby Monster,” Invent. Math. 144 (2001), 399-433. IVANOV AND WIEDORN
6. A.A. Ivanov and S.V. Shpectorov, “The flag-transitive tilde and Petersen-type geometrics are all known,” Bull. Amer. Math. Soc. New Ser. 31(2) (1994), 172-184.
7. A.A. Ivanov, “On geometries of the Fischer groups,” Eur. J. Comb. 16(2) (1995), 163-183.
8. A.A. Ivanov, Geometry of Sporadic Groups I. Peterson and Tilde Geometries, Cambridge University Press, Cambridge, 1999.
9. T. Meixner, “Some polar towers,” European J. Combin. 12 (1991), 397-451.
10. D.V. Pasechnik, “Geometric characterization of the sporadic groups Fi22, Fi23, and Fi* ,” J. Comb. Th. (A) 24 68 (1994), 100-114.
11. D.V. Pasechnik, “Extended polar spaces of rank at least 3,” J. Comb. Th. (A) 72(2) (1995), 232-242.
12. A. Pasini, Diagram Geometries, Oxford University Press, Oxford, 1994.
13. M. Ronan, “Embeddings and hyperplanes of discrete geometries,” European J. Combin. 8 (1987), 179-185.
14. G. Seitz, “Flag-transitive subgroups of Chevalley groups,” Ann. Math. 97 (1973), 27-56.
15. J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, 386, Springer, Berlin, 1974.