Calculation of Fundamental groups of $\rm II_1$ factors is, in general, an extremely important and hard problem. In this direction, a conjecture due to A. Connes states that the Fundamental group of the group von Neumann algebra $L(G)$, where $G$ is an i.c.c. property (T) group, is trivial.
In this talk, I shall provide the first examples of property (T) group factors with trivial Fundamental group. This talk is based on a recent joint work with Ionut Chifan, Cyril Houdayer and Krishnendu Khan.