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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)].

22F30 Homogeneous spaces
51A99 Linear incidence geometry
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