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

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.