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

In this article we provide the first examples of property (T) $\mathrm{I}{\mathrm{I}}_1$ factors $\mathcal{N}$ with trivial fundamental group, $\mathcal F (\mathcal N)=1$\mathrm{.}Ourexamplesariseasgroupfactors$\mathcal N=\mathcal L(G)$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.