Geometry and some of its relations within mathematics (Q2702038)

If \(\Gamma\) is a lattice in a noncompact, semisimple Lie group \(G\) with the property that every homomorphism \(\rho\colon \Gamma \to \widetilde G\) with Zariski dense image in a simple, noncompact Lie group \(\widetilde G\) with trivial center admits an extension to a homomorphism \(G \to \widetilde G\), then \(\Gamma\) is said to be superrigid.NEWLINENEWLINENEWLINEIn this survey, the author presents and makes comments on geometries of symmetric spaces by using fundamental groups and rigidity. The properties presented in the paper include the theorems by the author and \textit{A.~Candel} [Comment. Math. Helv. 74, No. 1, 63-83 (1999; Zbl 0943.53034)], \textit{G.~D.~Mostow} [Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies. No. 78. Princeton (1973; Zbl 0265.53039)], \textit{G.~A.~Margulis} [Proc. Int. Congr. Math., Vancouver 1974, Vol. 2, 21-34 (1975; Zbl 0336.57037)], \textit{W.~Ballmann} and \textit{P.~Eberlein} [J. Differ. Geom. 25, 1-22 (1987; Zbl 0701.53070)], \textit{K.~Corlette} [Ann. Math. (2) 135, No. 1, 165-182 (1992; Zbl 0768.53025)], and \textit{F.~Labourie} [Proc. Am. Math. Soc. 111, No. 3, 877-882 (1991; Zbl 0783.58016)].NEWLINENEWLINEFor the entire collection see [Zbl 0948.00009].