Hostname: page-component-7c8c6479df-ws8qp Total loading time: 0 Render date: 2024-03-29T15:23:19.401Z Has data issue: false hasContentIssue false

Application of the Bruhat-Tits tree of SU3(h) to some Ã2 groups

Published online by Cambridge University Press:  09 April 2009

Donald I. Cartwright
Affiliation:
School of Mathematics and Statistics The University of SydneyNSW 2006, Australia
Tim Steger
Affiliation:
Istituto di Matematica e Fisica Università di Sassarivia Vienna 207100 Sassari, Italy
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K be a nonarchimedean local field, let L be a separable quadratic extension of K, and let h denote a nondegenerate sesquilinear formk on L3. The Bruhat-Tits building associated with SU3(h) is a tree. This is applied to the study of certain groups acting simply transitively on vertices of the building associated with SL(3, F), F = Q3 or F3((X)).

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1998

References

[Br]Brown, K. S., Buildings (Springer, New York, 1989).Google Scholar
[CMSZ]Cartwright, D. I., Mantero, A. M., Steger, T. and Zappa, A., ‘Groups acting simply transitively on the vertices of a building of type Ā2’, I, II, Geom. Dedicata 47 (1993), 143166, 167–223.Google Scholar
[CMS]Carwright, D. I., Mlotowski, W. and Steger, T., ‘Property (T) and Ã2 groups’, Ann. Inst. Fourier 44 (1994), 213248Google Scholar
[Cas]Cassels, J. W. S., Local fields, London Math. Soc. Stud. Texts 3 (Cambridge University Press, Cambridge, 1986).Google Scholar
[Deu]Deuring, M., Algebren (Chelsea, New York, 1948).Google Scholar
[HV]de la Harpe, P. and Valette, A., ‘La Propriété (T) de Kazhdan pour les Groupes Localmente Compacts’, Astérisque, Soc. Math. France 174, (1989).Google Scholar
[Hum]Humphreys, J. E., Linear algebraic groups, Graduate Texts in Math. 45 (Springer, New York, 1975).Google Scholar
[Mar]Margulis, G. A., ‘Discrete subgroups of semisimple Lie groups’, Ergeb. Math. Grenzgeb. (3) 17 (Springer, Berlin, 1989).Google Scholar
[Ro]Ronan, M., Lectures on buildings, Perspect. Math. 7 (Academic Press, 1989).Google Scholar
[Ste]Steger, T., ‘Local fields and buildings’, Contemp. Math. 206 (1997), 79107.Google Scholar
[Ti]Tits, J., ‘Reductive groups over local fields’, Proc. Amer. Math. Soc. 33 (1979), 2969.Google Scholar
[We]Weil, A., Basic number theory, Die Grundlehren der math. Wissen., Band 144, (Springer, Berlin, 1974). (3rd Edition).Google Scholar