Every locally compact group is the outer automorphism group of a II₁ factor

Abstract We prove that every locally compact second countable group G arises as the outer automorphism group $\operatorname{Out} M$ 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 $\operatorname{Out} M$ .