Examples of property (T) II1 factors with trivial fundamental group

abstract:

In this article we provide the first examples of property (T) $\mathrm{II}_1$ 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 [Geom. Dedicata 112 (2005), 183--196] and the ones introduced recently in [Anal. PDE 16 (2023), 433--476] using the Belegradek-Osin Rips construction from [Groups Geom. Dyn. 2 (2008), 1--12]. In particular, our results provide a continuum of explicit pairwise nonisomorphic property (T) factors.