SOME APPLICATIONS OF GROUP-THEORETIC RIPS CONSTRUCTIONS TO THE CLASSIFICATION OF VON NEUMANN ALGEBRAS

We study various von Neumann algebraic rigidity aspects for the property (T) groups that arise via the Rips construction developed by Belegradek and Osin (Groups Geom. Dyn. 2:1 (2008), 1–12) in geometric group theory. Specifically, developing a new interplay between Popa’s deformation/rigidity theory (Int. Congr. Math, I (2007), 445–477) and geometric group theory methods, we show that several algebraic features of these groups are completely recognizable from the von Neumann algebraic structure. In particular, we obtain new infinite families of pairwise nonisomorphic property (T) group factors, thereby providing positive evidence towards Connes’ rigidity conjecture. In addition, we use the Rips construction to build examples of property (T) II1-factors which possess maximal von Neumann subalgebras without property (T), which answers a question raised by Y. Jiang and A. Skalski (arXiv:1903.08190 (2019), version 3).