Abstract
If a group Γ acts simply transitively on the vertices of an affine building Δ with connected diagram, then Δ must be of typeà n−1 for somen⩾2, and Γ must have a presentation of a simple type. The casen=2, when Δ is a tree, has been studied in detail. We consider the casen=3, motivated particularly by the case when Δ is the building ofG=PGL(3,K),K a local field, and when Γ⩽G. We exhibit such a group Γ whenK=F q ((X)),q any prime power. Our study leads to combinatorial objects which we calltriangle presentations. These triangle presentations give rise to some new buildings of typeà 2.
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References
Albert, A. A.,Structure of Algebras, Amer. Math. Soc. Colloq. Publications, Vol. XXIV, Amer. Math. Soc. 1939.
Betori, W. and Pagliacci, M., ‘Harmonic analysis for groups acting on trees’,Boll. Un. Mat. Ital. 3-B (1984), 333–349.
Brown, K. S.,Buildings, Springer-Verlag, 1989.
Figá-Talamanca, A. and Nebbia, C., ‘Harmonic analysis and representation theory for groups acting on homogeneous trees’,London Mathematical Society Lecture Note Series 162, Cambridge University Press, 1991.
Figá-Talamanca, Picardello, M. A., ‘Harmonic analysis on free groups’,Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1983.
Kantor, W. M., ‘Generalized polygons, SCABs and GABs’, inBuildings and the Geometry of Diagrams (L. A. Rosati, ed.),Lecture Notes in Math. 1181, Springer Verlag, 1970. pp. 79–158.
Köhler, P., Meixner, T. and Wester, M., ‘The affine building of typeà 2 over a local field of characteristic 2’,Archiv Math. 42 (1984), 400–407.
Margulis, G. A., ‘Discrete subgroups of semisimple Lie groups’,Ergeb. Math. Grenzgeb. (3), Band 17, Springer-Verlag, 1989.
Prasad, G., ‘Lattices in semisimple groups over local fields’, in ‘Studies in Algebra and Number Theory’ (G. Rota, ed.), Advances in Mathematics Supplementary Studies, Vol. 6, Academic Press, New York, 1979.
Ronan, M.,Lectures on Buildings, Academic Press, New York, 1989.
Serre, J.,Trees, Springer-Verlag, 1980.
Tits, J., ‘Sur le groupe des automorphismes d'un arbre’, inEssays on Topology and Related Topics, Memoires dédiés à G. de Rham, Springer-Verlag, 1970, pp. 188–211.
Tits, J., ‘Buildings and group amalgamations’, inProc. of Groups-St. Andrews, 1985, pp. 110–127 (E. F. Robertson and C. M. Campbell, eds);London Mathematical Society Lecture Note Series 121, Cambridge University Press, 1986.
Tits, J., ‘A local approach to buildings’, inThe Geometric Vein. The Coxeter Festschrift, Springer Verlag, New York, Heidelberg, Berlin, 1981 pp. 519–547.
Weil, A., ‘Basic number theory’,Grundlehren Math. Wiss. 144, Springer-Verlag, 1974.
Zimmer, R. J.,Ergodic Theory and Semisimple Groups, Birkhäuser, 1984.
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Cartwright, D.I., Mantero, A.M., Steger, T. et al. Groups acting simply transitively on the vertices of a building of typeà 2, I. Geom Dedicata 47, 143–166 (1993). https://doi.org/10.1007/BF01266617
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DOI: https://doi.org/10.1007/BF01266617