A bold conjecture due to Connes (1980) predicts that the group von Neumann algebra of an i.c.c. property (T) group completely remembers the group. The strong form of Connes' conjecture, due to Popa (2005) predicts that the factors arising from property (T) groups have trivial fundamental group. In this talk I shall discuss recent progress towards these conjectures, and present the first examples of property (T) factors with trivial fundamental group. This talk is based on a joint work with Ionut Chifan, Cyril Houdayer, and Krishnendu Khan.