In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group G can, under some circumstances, be as good as a representation of G itself. That this phenomenon happens for certain broadly defined classes of lattices inside semisimple groups was the discovery of Grigory Margulis, who proved some fundamental results in this direction.
There is more than one result that goes by the name of Margulis superrigidity. One simplified statement is this: take G to be a simply connected semisimple real algebraic group in GLn, such that the Lie group of its real points has real rank at least 2 and no compact factors. Suppose Γ is an irreducible lattice in G. For a local field F and ρ a linear representation of the lattice Γ of the Lie group, into GLn (F), assume the image ρ(Γ) is not relatively compact (in the topology arising from F) and such that its closure in the Zariski topology is connected. Then F is the real numbers or the complex numbers, and there is a rational representation of G giving rise to ρ by restriction.
- Margulis 1991, p. 2 Theorem 2.
- Hazewinkel, Michiel, ed. (2001) , "Discrete subgroup", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Lizhen JI, A summary of the work of Grigori Margulis, available at http://www.math.lsa.umich.edu/~lji/margulis.pdf, (see pages 17–19).
- Margulis, Grigory (1991). Discrete subgroups of semisimple Lie groups. Ergebnisse de Mathematik und ihrer Grenzgebiete. Springer-Verlag. pp. x+388. ISBN 3-540-12179-X. MR 1090825.
- (in French) Jacques Tits, Travaux de Margulis sur les sous-groupes discrets sur les groupes de Lie, Bourbaki Seminar 1975-6 exp. 482