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

We prove that every locally compact second countable group G arises as the outer automorphism group Out M of a II₁ 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 may all be realized as Out M and for which we prove several basic properties and characterizations.