Examples of property (T) II$_1$ factors with trivial fundamental group

TL;DR

This paper constructs the first known examples of property (T) II₁ factors with trivial fundamental group, using specific groups from existing literature, resulting in a continuum of non-isomorphic factors.

Contribution

It provides the first explicit examples of property (T) II₁ factors with trivial fundamental group, expanding the understanding of their structure and diversity.

Findings

First examples of property (T) II₁ factors with trivial fundamental group

Construction of a continuum of pairwise non-isomorphic property (T) factors

Use of groups from Valette and Belegradek-Osin Rips construction

Abstract

In this article we provide the first examples of property (T) $\rm II_1$ factors $\mathcal N$ with trivial fundamental group, $\mathcal F (\mathcal N)=1$. Our examples arise as group factors $\mathcal N=\mathcal L(G)$ where $G$ belong to two distinct families of property (T) groups previously studied in the literature: the groups introduced by Valette in \cite{Va04} and the ones introduced recently in \cite{CDK19} using the Belegradek-Osin Rips construction from \cite{BO06}. In particular, our results provide a continuum of explicit pairwise non-isomorphic property (T) factors.