Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Descent of affine buildings - II. Minimal angle $\pi /3$ and exceptional quadrangles
HTML articles powered by AMS MathViewer

by Koen Struyve PDF
Trans. Amer. Math. Soc. 366 (2014), 4367-4381 Request permission

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
  • 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 [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • 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, DOI 10.1007/978-3-662-12494-9
  • 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.
  • 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.
  • J. W. S. Cassels, Local fields, London Mathematical Society Student Texts, vol. 3, Cambridge University Press, Cambridge, 1986. MR 861410, DOI 10.1017/CBO9781139171885
  • 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, DOI 10.1090/coll/056
  • Norbert Knarr, Projectivities of generalized polygons, Ars Combin. 25 (1988), no. B, 265–275. Eleventh British Combinatorial Conference (London, 1987). MR 942482
  • B. Mühlherr, K. Struyve and H. Van Maldeghem, Descent of affine buildings - I. Large minimal angles, this issue.
  • 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, DOI 10.1090/conm/262/04180
  • Guy Rousseau, Immeubles des groupes réducitifs sur les corps locaux, Publications Mathématiques d’Orsay, No. 221-77.68, Université Paris XI, U.E.R. Mathématique, Orsay, 1977 (French). Thèse de doctorat. MR 0491992
  • O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. MR 0043776
  • A. Steinbach, Realizing Moufang quadrangles of type $\mathsf {E}_n$ inside Chevalley groups, preprint.
  • 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, pp. 33–62. MR 0224710
  • 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, DOI 10.1007/BFb0075514
  • Jacques Tits and Richard M. Weiss, Moufang polygons, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. MR 1938841, DOI 10.1007/978-3-662-04689-0
  • Richard M. Weiss, Quadrangular algebras, Mathematical Notes, vol. 46, Princeton University Press, Princeton, NJ, 2006. MR 2177056
  • Richard M. Weiss, The structure of affine buildings, Annals of Mathematics Studies, vol. 168, Princeton University Press, Princeton, NJ, 2009. MR 2468338
  • Richard M. Weiss, On the existence of certain affine buildings of type $E_6$ and $E_7$, J. Reine Angew. Math. 653 (2011), 135–147. MR 2794628, DOI 10.1515/CRELLE.2011.022
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)
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 366 (2014), 4367-4381
  • MSC (2010): Primary 51E24, 20E42
  • DOI: https://doi.org/10.1090/S0002-9947-2014-05986-2
  • MathSciNet review: 3206463