Automatic continuity of homomorphisms and fixed points on metric compacta

摘要

We prove that arbitrary homomorphisms from one of the groups Homeo(2(N)), Homeo(2(N))(N), Aut(Q, <), Homeo(R) or Homeo(S-1) into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination with a result of V. G. Pestov, that any action of the discrete group Homeo(+)(R) by homeomorphisms on a compact metric space has a fixed point.