Universal Covers of Geometries of Far Away Type
Antonio Pasini
DOI: 10.1023/B:JACO.0000011938.63918.69
Abstract
The geometries studied in this paper are obtained from buildings of spherical type by removing all chambers at non-maximal distance from a given element or flag. I consider a number of special cases of the above construction chosen among those which most frequently appear in the literature, proving that the resulting geometry is always simply connected but for three cases of small rank defined over GF(2) and GF(4). I also compute the universal cover in those exceptional cases.
Pages: 211–243
Keywords: buildings; universal covers; embeddings; binary codes
Full Text: PDF
References
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2. B. Baumeister, T. Meixner, and A. Pasini, “GF(2)-expansions,” Geom. Dedicata 67 (1997), 163-180.
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8. H. Cuypers and A. Pasini, “Locally polar geometries with affine planes,” European J. Combin. 13 (1992), 39-57.
9. C. Huybrechts and A. Pasini, “Flag-transitive extensions of dual projective spaces,” Bull. Soc. Math. Belgique 5 (1998), 341-352.
10. A. Munemasa and S.V. Shpectorov, “A local characterization of the graphs of alternating forms,” in Finite Geometry and Combinatorics, F. De Clerck et al. (Eds.), Cambridge Univ. Press, 1993, pp. 289-302.
11. A. Munemasa, D.V. Pasechnik, and S.V. Shpectorov, “A local characterization of the graphs of alternating forms and the graphs of the quadratic forms over GF(2),” in Finite Geometry and Combinatorics, F. De Clerck et al. (Eds.), Cambridge Univ. Press, 1993, pp. 303-317.
12. A. Pasini, “On locally polar geometries whose planes are affine,” Geom. Dedicata 34 (1990), 35-56.
13. A. Pasini, Diagram Geometry, Oxford Univ. Press, 1994.
14. A. Pasini, “Shadow geometries and simple connectedness,” European J. Combin. 15 (1994), 17-34.
15. A. Pasini, “Gluing two affine spaces,” Bull. Belgian Math. Soc. 3 (1996), 25-40.
16. A. Pasini, “Embeddings and expansions,” Bull. Belgian Math. Soc. to appear.
17. S. Rinauro, “On some extensions of generalized quadrangles of grid type,” J. Geometry 38 (1990), 158-164.
18. J. Tits, Buildings of Spherical Type and Finite BN-pairs, Springer Lect. Notes 1974, vol. 386.
19. L. Van Nypelseer, “Rank n geometries with affine hyperplanes and dual affine point residues,” European J. Combin. 12 (1991), 561-566.
2. B. Baumeister, T. Meixner, and A. Pasini, “GF(2)-expansions,” Geom. Dedicata 67 (1997), 163-180.
3. B. Baumeister, S. Shpectorov, and G. Stroth, “Flag-transitive affine dual polar spaces,” preprint, July 1997.
4. R.J. Blok and A.E. Brouwer, “The geometry far from a residue,” in Groups and Geometries, L. Di Martino et al. (Eds.), Birkh\ddot auser, 1998, pp. 29-38.
5. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance Regular Graphs, Springer, Berlin, 1989.
6. F. Buekenhout, Handbook of Incidence Geometry, Elsevier, Amsterdam, 1995.
7. A. Cohen and E. Shult, “Affine polar spaces,” Geom. Dedicata 35 (1990), 43-76.
8. H. Cuypers and A. Pasini, “Locally polar geometries with affine planes,” European J. Combin. 13 (1992), 39-57.
9. C. Huybrechts and A. Pasini, “Flag-transitive extensions of dual projective spaces,” Bull. Soc. Math. Belgique 5 (1998), 341-352.
10. A. Munemasa and S.V. Shpectorov, “A local characterization of the graphs of alternating forms,” in Finite Geometry and Combinatorics, F. De Clerck et al. (Eds.), Cambridge Univ. Press, 1993, pp. 289-302.
11. A. Munemasa, D.V. Pasechnik, and S.V. Shpectorov, “A local characterization of the graphs of alternating forms and the graphs of the quadratic forms over GF(2),” in Finite Geometry and Combinatorics, F. De Clerck et al. (Eds.), Cambridge Univ. Press, 1993, pp. 303-317.
12. A. Pasini, “On locally polar geometries whose planes are affine,” Geom. Dedicata 34 (1990), 35-56.
13. A. Pasini, Diagram Geometry, Oxford Univ. Press, 1994.
14. A. Pasini, “Shadow geometries and simple connectedness,” European J. Combin. 15 (1994), 17-34.
15. A. Pasini, “Gluing two affine spaces,” Bull. Belgian Math. Soc. 3 (1996), 25-40.
16. A. Pasini, “Embeddings and expansions,” Bull. Belgian Math. Soc. to appear.
17. S. Rinauro, “On some extensions of generalized quadrangles of grid type,” J. Geometry 38 (1990), 158-164.
18. J. Tits, Buildings of Spherical Type and Finite BN-pairs, Springer Lect. Notes 1974, vol. 386.
19. L. Van Nypelseer, “Rank n geometries with affine hyperplanes and dual affine point residues,” European J. Combin. 12 (1991), 561-566.