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

TL;DR

This paper explores the rigidity properties of von Neumann algebras associated with property (T) groups constructed via Rips methods, revealing new non-isomorphic factors and subalgebra structures that support Connes' Rigidity Conjecture.

Contribution

It introduces novel techniques combining geometric group theory and deformation/rigidity theory to classify von Neumann algebras from Rips construction groups.

Findings

Produced new families of non-isomorphic property (T) group factors

Constructed property (T) II$_1$ factors with maximal subalgebras lacking property (T)

Provided evidence supporting Connes' Rigidity Conjecture

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.