  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 $M$.
