Generalizing the Margulis Normal Subgroup Theorem, Margulis and Zimmer conjectured that any subgroup of a lattice in a higher-rank Lie group which is commensurated by the lattice is (up to finite index) of a standard form. I will present some of my work on property (T) for totally disconnected groups and countable dense subgroups and explain how it provides "half" of the solution to the conjecture. This is joint work with Yehuda Shalom.