Some Applications of Group Theoretic Rips Constructions to the Classification of von Neumann Algebras

Abstract

In this paper we study various von Neumann algebraic rigidity aspects for the property (T) groups that arise via the Rips construction developed by Belegradek and Osin in geometric group theory \cite{BO06}. Specifically, developing a new interplay between Popa's deformation/rigidity theory \cite{Po07} 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 non-isomorphic 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${}_1$ factors which posses maximal von Neumann subalgebras without property (T) which answers a question raised in an earlier version of \cite{JS19} by Y. Jiang and A. Skalski.

arXiv commentFinal version. To appear in Analysis and PDE

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• Maximal subgroups and von Neumann subalgebras with the Haagerup property
• Product rigidity in von Neumann and C${}^{\ast }$-algebras via s-malleable deformations
• New examples of Property (T) factors with trivial fundamental group and unique prime factorization
• Wreath-like products of groups and their von Neumann algebras I: $W^{\ast }$-superrigidity
• Rigidity for von Neumann algebras of graph product groups. I. Structure of automorphisms