Rigidity of lattices: An introduction. (English) Zbl 0786.22015

Geometric topology: recent developments, Lect. 1st Sess. CIME, Montecatini Terme/Italy 1990, Lect. Notes Math. 1504, 39-137 (1991).
[For the entire collection see Zbl 0746.00065.]
These are a set of lectures by the authors on the topic of the title.
The strong rigidity theorem of Mostow [see G. D. Mostow, Strong rigidity of symmetric spaces (Ann. Math. Stud. 78) (Princeton 1973; Zbl 0265.53039)] and some ideas of Furstenberg on ideal boundaries [H. Furstenberg, Boundary theory and stochastic processes on homogeneous spaces, Proc. Symp. Pure Math. 26, 193-229 (1973; Zbl 0289.22011)] are discussed. Then follows a detailed description of the super-rigidity of lattices in simple Lie groups of \(\mathbb{R}\text{-rank}\geq 2\), proved by Margulis [see G. A. Margulis, Discrete subgroups of semisimple Lie groups (Springer 1990; Zbl 0732.22008)].
In the latter part of the lectures, the super-rigidity theorem of Corlette for lattices in \(Sp(n,1)\) and the \(\mathbb{R}\)-rank 1 form of \(F_ 4\) is presented [see K. Corlette, Ann. Math., II. Ser. 135, No. 1, 165-182 (1992; Zbl 0768.53025)]. It is to be mentioned that a \(p\)-adic version of Corlette’s theorem was proved by Gromov and Schoen, which establishes the arithmeticity of these lattices in \(Sp(n,1)\) and \(F_ 4\).


22E40 Discrete subgroups of Lie groups