Annals of Combinatorics 4 (2000) 125-137

On Opposition in Spherical Buildings and Twin Buildings

Peter Abramenko and Hendrik Van Maldeghem

Fakultät für 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.

Get the DVI | PS | PDF file of this abstract.