Every locally compact group is the outer automorphism group of a II$_1$ factor

We prove that every locally compact second countable group $G$ arises as the outer automorphism group Out $M$ of a II$_1$ factor, which was so far only known for totally disconnected groups, compact groups and a few isolated examples. We obtain this result by proving that every locally compact second countable group is a centralizer group, a class of Polish groups that arise naturally in ergodic theory and that may all be realized as Out $M$.