Abstract
We show that a 3-spherical building in which each rank 2 residue is connected far away from a chamber, and each rank 3 residue is simply 2-connected far away from a chamber, admits a twinning (i.e., is one half of a twin building) as soon as it admits a codistance, i.e., a twinning with a single chamber.
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Abramenko, P.: Twin buildings and applications to S-arithmetic groups. In: Lecture Notes in Mathematics, vol. 1641. Springer, Berlin (1996)
Abramenko, P., Brown, K.: Buildings, theory and applications. In: Graduate Texts in Mathematics, vol. 248. Springer, New York (2008)
Abramson M., Bennett C.: Embeddings of twin trees. Geometriae Dedicata 75(2), 209–215 (1999)
Brouwer, A.E.: The complement of a geometric hyperplane in a generalized polygon is usually connected. In: De Clerck, F., et al. (eds.) Finite Geometry and Combinatorics. Cambridge University Press, Cambridge (1993)
Devillers A., Mühlherr B.: On the simple connectedness of certain subsets of buildings. Forum Math. 19(6), 955–970 (2007)
Dress A., Scharlau R.: Gated sets in metric spaces. Aequationes Math. 34, 112–120 (1987)
Humphreys, J.E.: Reflection groups and Coxeter groups. In: Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)
Mühlherr, B.: Coxeter groups in Coxeter groups. In: De Clerck, F., et al. (eds.) Finite Geometries. Cambridge University Press, Cambridge (1992)
Mühlherr B.: A rank 2 characterization of twinnings. Eur. J. Combin. 19(5), 603–612 (1998)
Mühlherr, B.: On the Existence of 2-Spherical Twin Buildings, Habilitationsschrift, Universität Dortmund (1999)
Mühlherr B.: Locally split and locally finite twin buildings of 2-spherical type. J. Reine Angew. Math. 511, 119–143 (1999)
Mühlherr B., Van Maldeghem H.: Codistances in buildings. Innov. Incid. Geom. 10, 81–91 (2010)
Ronan M.: Lectures on Buildings. Academic Press, San Diego (1989)
Ronan M.: A local approach to twin buildings, preprint (2008)
Ronan M., Tits J.: Building buildings. Math. Ann. 278(1–4), 291–306 (1987)
Ronan M., Tits J.: Twin trees II: local structure and a universal construction. Israel J. Math. 109, 349–377 (1999)
Tits, J.: Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, vol. 386. Springer, Berlin (1974)
Tits, J.: A local approach to buildings. In: Davis, C. (ed.) Coxeter Festschrift. The geometric vein, pp. 519–547. Springer, New York (1981)
Tits J.: Ensembles ordonnés, immeubles et sommes amalgamées. Bull. Soc. Math. Belg. Sér. A 38, 367–387 (1986)
Tits J.: Immeubles de type affine. In: Buildings and the geometry of diagrams (Como, 1984). Lecture Notes in Math., vol. 1181, pp. 159–90. Springer, Berlin (1986)
Tits, J.: Twin buildings and groups of Kac-Moody type. In: Groups, Combinatorics and Geometry (Durham, 1990) (London Math. Soc. Lecture Note Ser., 165), pp. 249–286. Cambridge University Press, Cambridge (1992)
Weiss R.: The Structure of Spherical Buildings. Princeton University Press, Princeton (2003)
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Most of the work for this paper was done while A. Devillers was Collaborateur Scientifique of the Fonds National de la Recherche Scientifique (Belgium).
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Devillers, A., Mühlherr, B. & Van Maldeghem, H. Codistances of 3-spherical buildings. Math. Ann. 354, 297–329 (2012). https://doi.org/10.1007/s00208-011-0733-5
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DOI: https://doi.org/10.1007/s00208-011-0733-5