  The notion of strong 1-boundedness for finite von Neumann algebras was
introduced by Jung in arXiv:math/0510576 . This framework provided a free
probabilistic approach to study rigidity properties and classification of
finite von Neumann algebras. In this paper, we prove that tracial von Neumann
algebras with a finite Kazhdan set are strongly 1-bounded. This includes all
Property (T) von Neumann algebras with finite dimensional center and group von
Neumann algebras of Property (T) groups. This result generalizes all the
previous results in this direction due to Voiculescu, Ge, Ge-Shen,
Connes-Shlyakhtenko, Jung-Shlyakhtenko, Jung, and Shlyakhtenko. Our proofs are
based on analysis of covering estimates of microstate spaces using an iteration
technique in the spirit of Jung.
