We study various von Neumann algebraic rigidity aspects for the property (T) groups
that arise via the Rips construction developed by Belegradek and Osin (GroupsGeom. 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)
II-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).