On quasi-homomorphism rigidity for lattices in simple algebraic groups

Property (TTT) was introduced by Ozawa as a strengthening of Kazhdan’s property (T) and Burger and Monod’s property (TT) . In this paper, we improve Ozawa’s result by showing that any simple algebraic group of rank \geq 2 over a local field has property (TTT) . We also show that lattices in a locally compact second countable group inherit property (TTT) . Finally, we study to what extent Lie groups with infinite center fail to have properties (TT) and (TTT) .