Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Descent of affine buildings - II. Minimal angle $ \pi/3$ and exceptional quadrangles

Author: Koen Struyve
Journal: Trans. Amer. Math. Soc. 366 (2014), 4367-4381
MSC (2010): Primary 51E24, 20E42
Published electronically: April 16, 2014
MathSciNet review: 3206463
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this two-part paper we prove an existence result for affine buildings arising from exceptional algebraic reductive groups. Combined with earlier results on classical groups, this gives a complete and positive answer to the conjecture concerning the existence of affine buildings arising from such groups defined over a (skew) field with a complete valuation, as proposed by Jacques Tits.

This second part builds upon the results of the first part and deals with the remaining cases.

References [Enhancements On Off] (What's this?)

  • [1] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • [2] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
  • [3] F. Bruhat and J. Tits, Groupes réductifs sur un corps local: I. Données radicielles valuées, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5-252.
  • [4] F. Bruhat and J. Tits, Groupes réductifs sur un corps local: II. Schémas en groupes. Existence d'une donnée racidielle valuée, Inst. Hautes Études Sci. Publ.Math. 60 (1984), 5-184.
  • [5] J. W. S. Cassels, Local fields, London Mathematical Society Student Texts, vol. 3, Cambridge University Press, Cambridge, 1986. MR 861410
  • [6] Richard Elman, Nikita Karpenko, and Alexander Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56, American Mathematical Society, Providence, RI, 2008. MR 2427530
  • [7] Norbert Knarr, Projectivities of generalized polygons, Ars Combin. 25 (1988), no. B, 265–275. Eleventh British Combinatorial Conference (London, 1987). MR 942482
  • [8] B. Mühlherr, K. Struyve and H. Van Maldeghem, Descent of affine buildings - I. Large minimal angles, this issue.
  • [9] Anne Parreau, Immeubles affines: construction par les normes et étude des isométries, Crystallographic groups and their generalizations (Kortrijk, 1999) Contemp. Math., vol. 262, Amer. Math. Soc., Providence, RI, 2000, pp. 263–302 (French, with English summary). MR 1796138,
  • [10] Guy Rousseau, Immeubles des groupes réducitifs sur les corps locaux, U.E.R. Mathématique, Université Paris XI, Orsay, 1977 (French). Thèse de doctorat; Publications Mathématiques d’Orsay, No. 221-77.68. MR 0491992
  • [11] O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. MR 0043776
  • [12] A. Steinbach, Realizing Moufang quadrangles of type $ \mathsf {E}_n$ inside Chevalley groups, preprint.
  • [13] J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, 1966, pp. 33–62. MR 0224710
  • [14] Jacques Tits, Immeubles de type affine, Buildings and the geometry of diagrams (Como, 1984) Lecture Notes in Math., vol. 1181, Springer, Berlin, 1986, pp. 159–190 (French). MR 843391,
  • [15] Jacques Tits and Richard M. Weiss, Moufang polygons, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. MR 1938841
  • [16] Richard M. Weiss, Quadrangular algebras, Mathematical Notes, vol. 46, Princeton University Press, Princeton, NJ, 2006. MR 2177056
  • [17] Richard M. Weiss, The structure of affine buildings, Annals of Mathematics Studies, vol. 168, Princeton University Press, Princeton, NJ, 2009. MR 2468338
  • [18] Richard M. Weiss, On the existence of certain affine buildings of type 𝐸₆ and 𝐸₇, J. Reine Angew. Math. 653 (2011), 135–147. MR 2794628,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 51E24, 20E42

Retrieve articles in all journals with MSC (2010): 51E24, 20E42

Additional Information

Koen Struyve
Affiliation: Department of Pure Mathematics, Ghent University, B-9000 Ghent, Belgium

Received by editor(s): January 19, 2012
Received by editor(s) in revised form: September 23, 2012, and October 15, 2012
Published electronically: April 16, 2014
Additional Notes: The author was supported by the Fund for Scientific Research – Flanders (FWO - Vlaanderen)
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.