Archimedean superrigidity and hyperbolic geometry.(English)Zbl 0768.53025

The famous “superrigidity theorem” of G. A. Margulis [Am. Math. Soc., Transl., II. Ser. 109, 33-45 (1977; Zbl 0367.57012)] for lattices in semisimple Lie groups of real rank at least two (simultaneously providing arithmeticity of all such lattices) left unanswered the question of whether lattices in groups of real rank one are superrigid. Here there are essentially four cases to consider: the automorphism groups of real, complex and quaternionic hyperbolic spaces together with that of the hyperbolic Cayley plane (we refer to archimedean fields: $$p$$- adic issue is not considered). In contrast to the first two cases where non-arithmetic lattices exist [see first examples in V. S. Makarov, Sov. Math. Dokl. 7, 328-331 (1966); translation from Dokl. Akad. Nauk SSSR 167, 30-33 (1966; Zbl 0146.165) and G. Mostow, Pac. J. Math. 86, 171-276 (1980; Zbl 0456.22012)] and the superrigidity over archimedean fields does not hold [for first examples, see B. Apanasov, Ann. Math. Stud. 97, 21-31 (1981; Zbl 0464.30037)], the author shows that superrigidity over archimedean fields holds for lattices in the remaining two cases.

MSC:

 53C35 Differential geometry of symmetric spaces 57S25 Groups acting on specific manifolds 22E40 Discrete subgroups of Lie groups
Full Text: