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Some applications of group-theoretic Rips constructions to the classification of von Neumann algebras

Ionuţ Chifan, Sayan Das and Krishnendu Khan

Vol. 16 (2023), No. 2, 433–476

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).

Mathematical Subject Classification
Primary: 20F06, 46L36
Secondary: 20F65, 20F67
Received: 2 June 2020
Revised: 8 July 2021
Accepted: 11 August 2021
Published: 3 May 2023
Ionuţ Chifan
Department of Mathematics
The University of Iowa
Iowa City, IA
United States
Sayan Das
Department of Mathematics
University of California Riverside
Riverside, CA
United States
Krishnendu Khan
Department of Mathematics
The University of Iowa
Iowa City, IA
United States

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