Property (t ) and Rigidity for Actions on Banach Spaces

1.a. Since its introduction by Kazhdan in [Ka], property (T ) became a fundamental concept in mathematics with a wide range of applications to such areas as: • The structure of infinite groups—finite generation and finite Abelianization of higher-rank lattices [Ka], obstruction to free or amalgamated splittings [Wa], [A], [M4], structure of normal subgroups [M2] etc.; • Combinatorics—the first construction of expanders [M1] (see [Lu]); • Operator algebras—factors of type II1 whose fundamental group is countable [C] or even trivial [Po1]; rigidity theorems for the factors associated to the Kazhdan group [Po2]; • Ergodic theory—rigidity results related to orbit equivalence [Po3], [Hj]; the Banach–Ruziewicz problem [M3], [Su]; • Smooth dynamics—local rigidity [FM1], [FM2]; actions on the circle [N1] (and [PS], [Rz]). It has also been an important tool in providing interesting (counter)examples: to Day’s “von Neumann conjecture” [Gr1, §5.6] and in the context of the Baum–Connes conjecture [HLS] (related to [Gr2]).