Automatic continuity of homomorphisms and fixed points on metric compacta

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We prove that arbitrary homomorphisms from one of the groups Homeo(), Homeo(), Aut(,<), Homeo(), or Homeo(S¹) 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 on V.G. Pestov, that any action of the discrete group Homeo+() by homeomorphisms on a compact metric space has a fixed point.