Semidirect product rigidity of group von Neumann algebras arising from class $\mathscr{S}$, inductive limits and fundamental group

TL;DR

This paper investigates the rigidity properties of group von Neumann algebras derived from property (T) groups, especially those from Rips constructions and inductive limits, demonstrating new examples with trivial fundamental groups and Cartan rigidity.

Contribution

It extends the class of property (T) groups whose von Neumann algebras retain semidirect product features and produces new examples of factors with trivial fundamental group, including inductive limit groups.

Findings

Produced new property (T) group factors with trivial fundamental group.

Extended the class of groups preserving semidirect product features in von Neumann algebras.

Established Cartan rigidity for these groups.

Abstract

In this article we study property (T) groups arising from Rips construction in geometric group theory in the spirit of \cite{CDK19} and certain inductive limit groups from this class. Using interplay between Popa's deformation/rigidity and methods in geometric group theory we are able to extend the class of groups considered in \cite{CDK19} that remembers semidirect product features while passing to the group von Neumann algebras. Combining these results with the method developed in \cite{CDHK20} we are able to produce more examples of property (T) group factors with trivial fundamental group. The inductive limit groups do not have property (T) and provides examples of more factors with trivial fundamental group. We are also able to show Cartan rigidity for these groups.