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

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