Annals of Combinatorics 4 (2000) 125-137
On Opposition in Spherical Buildings and Twin Buildings
Peter Abramenko and Hendrik Van Maldeghem
Mathematik, Universität Bielefeld, Postfach
100131, D-33501 Bielefeld, Germany
Department of Pure Mathematics and Computer Algebra,
University of Ghent, Galglaan 2, B-9000 Ghent, Belgium
Received February 20, 1999
AMS Subject Classification: 51E24
Abstract. In this paper, we prove a combinatorial property of twin apartments and opposition of chambers in twin buildings. We then characterize adjacency of chambers in twin buildings by means of opposition of chambers. As an application, we study maps which satisfy certain conditions related to opposition of chambers, e.g., maps that preserve opposition. Applied to the special case of spherical buildings, all our main results as well as their corollaries are new.
Keywords: twin buildings, spherical buildings, opposition, apartments, generalized polygons, chambers
1. P. Abramenko, Twin Buildings and Applications to S-Arithmetic Groups, Lecture Notes in Mathematics, Vol. 1641, Springer-Verlag, 1996.
2. B. M¨uhlherr, A rank 2 characterization of twinnings, European J. Combin. 19 (1998) 603– 612.
3. J. Tits , Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, Vol. 386, Springer, Berlin–Heidelberg–New York, 1974.
4. J. Tits, Résumé de cours, Annuaire du Collège de France, 89e année (1988-89), 81–95.
5. J. Tits, Twin buildings and groups of Kac–Moody type, In: Proceedings on Groups, Combinatorics and Geometry, Durham, 1990, M. Liebeck and J. Saxl, Eds., Cambridge University Press, 1992, pp. 249–286.
6. H. Van Maldeghem, Generalized Polygons, Monographs in Mathematics, Vol. 93, Birkh¨auser- Verlag, Basel, 1998.