Since Shalom's breakthrough of Bounded generations and Kazhdan's property (T) in 1999, it had been a big problem to remove any form of "bounded generation" condition from algebraization processes in proving fixed point property. In this talk, we provide the affirmative resolution of this problem (arXiv:1505.06728).
Applications of our main theorem ("strong algebraization") include non-commutative universal lattices, and (Kac-Moody-)Steinberg groups over finitely generated unital ring.