Abstract
Let Γ be a finite geometry of rank n ≥ 2 with a selected type of elements, called'points'. Let m be the number of'points' of Γ. Under some mild hypotheses on Γ we can consider an affine expansion of Γ to AG(m,2). We prove that the geometries obtained by applying this construction to matroids are simply connected. Then we exploit this result to study universal covers of certain geometries arising from hyperbolic quadrics and symplectic varieties over GF.
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References
Baumeister B. and Pasini, A.: On flat flag-transitive c:c _-geometries, J. Algebraic Combin. 6 (1997), 5–26.
Brouwer, A., Cohen, A. and Neumaier, A.: Distance-Regular Graphs, Springer-Verlag, New York, 1989.
Buekenhout, F., Dehon, M. and De Schutter, I: Projective injection of geometries and their affine extensions, J. Geometry 52 (1995), 41–53.
Buekenhout, F., Huybrechts, C. and Pasini, A.: Parallelismin diagram geometry, Bull. Soc. Math. Belgique–Simon Stevin 1 (1994), 355–397.
Munemasa, A., Pasechnik, D. and Shpectorov, S.: A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2), in F. De Clerck et al. (eds), Finite Geometry and Combinatorics (Astene–Deinze, 1992), Cambridge Univ. Press, 1993, pp. 303–317.
Pasini, A.: Covers of finite geometries with non-spherical minimal circuit diagram, in Buildings and the Geometry of Diagrams, Lecture Notes in Math. 1181, Springer-Verlag, New York, 1986, pp. 218–241.
Pasini, A.: Gluing two affine spaces, Bull. Soc. Math. Belgique 3 (1996), 25–40.
Pasini, A.: Diagram Geometries, Oxford University Press, Oxford, 1994.
Shpectorov, S.: private communication.
Shult, E.: Geometric hyperplanes of the half-spin geometries arise from embeddings, Bull. Soc. Math. Belgique–Simon Stevin 1 (1994), 439–453.
Shult, E. and Thas, J.: Hyperplanes of dual polar spaces and the spin module, Arch. Math. 59 (1992), 610–623.
Shult, E. and Yanushka, A.: Near n-gons and line systems, Geom. Dedicata 9 (1980), 1–72.
Yoshiara, S. and Pasini, A.: Generalized towers of flag-transitive circular extensions of a non-classical C3-geometry, J. Combin. Theory Ser. A 66 (1994), 313–320.
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Baumeister, B., Meixner, T. & Pasini, A. FG(2)-Expansions. Geometriae Dedicata 67, 163–180 (1997). https://doi.org/10.1023/A:1004913528398
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DOI: https://doi.org/10.1023/A:1004913528398