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Two-transitive Lie groups. (English) Zbl 1044.22014
In this excellently written paper, the author gives a new and complete proof for J. Tits’s classification of doubly transitive Lie group actions [Acad. Roy. Belgique, Cl. Sci., Mém., Coll. \(8^\circ 29\), No. 3 (1955; Zbl 0067.12301)]. The insight gained from his original proof had led Tits subsequently to develop the theory of buildings. The present author uses this theory, among other things, in his proof. In the case of a compact space (where the group is simple), he employs a characterisation of parabolic subgroups due to H. Furstenberg [Semin. Bourbaki, 32e année, vol. 1979/80, Exp. No. 559, Lect. Notes Math. 842, 273–292 (1981; Zbl 0471.22007)] in order to obtain a spherical building whenever the stabilizer is of real rank \(\geq 2\). He then proves that only spherical buildings of type \(A_k\), \(k\leq \omega\), admit a group acting 2-transitively on the residues of a fixed type. For the given action, this means that it is a projective linear action on some projective space. The rank one case leads to projective quadrics. If the space is non-compact, the author shows using some algebraic topology that it is a Euclidean space and the action is linear. It is then a matter of representation theory to determine the possibilities. The author derives from the complete classification the sub-classifications of all sharply 2-transitive actions (Kalscheuer, Tits, Grundhöfer), and of all complex algebraic 2-transitive actions [F. Knop, Arch. Math. 41, 438–446 (1983; Zbl 0557.14028)].

MSC:
22F30 Homogeneous spaces
51A99 Linear incidence geometry
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References:
[1] A. L. Besse, Einstein manifolds, Springer-Verlag, Berlin1987. · Zbl 0613.53001
[2] A. Borel, Some remarks about Lie groups transitive on spheres and tori, Bull. A.M.S. 55 (1949), 580-587. · Zbl 0034.01603
[3] R. Acad. Sci. Paris 230 pp 1378– (1950)
[4] Ann. Math. 58 pp 443– (1953)
[5] N. Bourbaki, Groupes et algebres de Lie, Ch. 4-6, Hermann, Paris 1968.
[6] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London 1972. · Zbl 0246.57017
[7] G. E. Bredon, Topology and geometry, Springer-Verlag, New York 1993. · Zbl 0791.55001
[8] A. E. Brouwer, A. Cohen, and A. Neumaier, Distance-regular graphs, Springer-Verlag, Berlin1989. · Zbl 0747.05073
[9] K. Burns and R. Spatzier, On topologicalTits buildings and their classification, Publ. Math. I.H.E.S. 65 (1987), 5-34. · Zbl 0643.53036
[10] B., Canad. J. Math. 28 pp 1021– (1976)
[11] J. D. Dixon, B. Mortimer, Permutation groups, Springer-Verlag, New York1996.
[12] Dress A., Aequat. Math. 34 pp 112– (1987)
[13] Eilenberg S., Amer. J. Math. 63 pp 39– (1941)
[14] Sem. Bourbaki 1979 pp 543–
[15] V. V. Gorbatsevich, A. L. Onishchik, and E. B. Vinberg, Foundations ofLie theory and Lie transformation groups, Springer-Verlag, Berlin 1997. · Zbl 0999.17500
[16] Forum Math. 1 pp 81– (1981)
[17] T. Grundh fer and R. L wen, Linear topological geometries, in: Handbook of incidence geometry, F. Buekenhout, ed., North-Holland, Amsterdam (1995), 1253-1324.
[18] Grundhfer T., Geom. Dedic. 83 pp 1– (2000)
[19] A. Hahn and T. O’Meara, The classical groups and K-theory, Springer-Verlag, Berlin 1989.
[20] S. Helgason, Di erential geometry, Lie groups, and symmetric spaces, Academic Press, Inc., New York-London 1978. · Zbl 0451.53038
[21] E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol. I, Springer-Verlag, Berlin-G ttingen-Heidelberg 1963. · Zbl 0115.10603
[22] J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, Cambridge 1990. · Zbl 0725.20028
[23] D. Husemoller, Fibre bundles, third edition, Springer-Verlag, New York1994. · Zbl 0202.22903
[24] N. Jacobson, Basic algebra, I, second edition, Freeman, New York1985. · Zbl 0284.16001
[25] Kalscheuer F., Abh. Math. Sem. Hamburg 13 pp 413– (1940)
[26] Knop F., Arch. Math. 41 pp 438– (1983)
[27] Kolmogorov A. N., Ann. Math. 33 pp 175– (1932)
[28] Kramer L., Mem. Amer. Math. Soc. 158 pp 752– (2002)
[29] L. Kramer, Buildings and classical groups, in: Tits buildings and the model theory of groups, K. Tent, ed., Cambridge Univ. Press, Cambridge (2002), 59-101. · Zbl 1040.51016
[30] Khne R., Abh. Math. Sem. Univ. Hamburg 62 pp 1– (1992)
[31] Montgomery D., Ann. Math. 44 pp 454– (1943)
[32] A. L. Onishchik, Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, Leipzig 1994. · Zbl 0796.57001
[33] A. L. Onishchik, E. B. Vinberg, Lie groups and algebraic groups, Springer-Verlag, Berlin 1990. · Zbl 0722.22004
[34] Comment. Math. Helv. 33 pp 109– (1959)
[35] M. Ronan, Lectures on buildings, Academic Press, Inc., Boston, MA, 1989. · Zbl 0694.51001
[36] H. Salzmann, D. Betten, T. Grundh fer, H. H hl, R. L wen, M. Stroppel, Compact projective planes. With an introduction to octonion geometry, De Gruyter Co., Berlin 1995. · Zbl 0851.51003
[37] R. Scharlau, Buildings, in: Handbook of incidence geometry, F. Buekenhout, ed., North-Holland, Amsterdam (1995), 477-645. · Zbl 0841.51005
[38] E. H. Spanier, Algebraic topology, Springer-Verlag, New York1989. · Zbl 0145.43303
[39] J. Fac. Sci. Univ. Tokyo Sect. I 10 pp 88– (1964)
[40] Comment. Math. Helv. 26 pp 203– (1952)
[41] Tits J., Fasc. pp 3– (1955)
[42] Proc. Symp. Pure Math. 9 pp 33– (1966)
[43] J. Tits, Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, Springer-Verlag, Berlin-New York 1967. · Zbl 0166.29703
[44] Tits J., Math. 247 pp 196– (1971)
[45] J. Tits, A local approach to buildings, in: The geometric vein The Coxeter Festschrift, Davis et al. eds., Springer Verlag, Berlin (1981), 519-547.
[46] J. Tits, Twin buildings and groups of Kac-Moody type, in: Groups, combinatorics and geometry (Durham 1990), M. Liebeck and J. Saxl ed., London Math. Soc. Lect. Notes 165, Cambridge Univ. Press, Cambridge (1992), 249-286. · Zbl 0851.22023
[47] Vlklein H., Arch. Math. 36 pp 23– (1981)
[48] G. Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, New York-Heidelberg 1972. · Zbl 0265.22020
[49] J. A. Wolf, Spaces of constant curvature, fifth edition, Publish or Perish, Inc., Houston, TX, 1984.
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