Automatic continuity of homomorphisms and fixed points on metric compacta

Abstract

We prove that arbitrary homomorphisms from one of the groups ${\rm Homeo}(\ca)$, ${\rm Homeo}(\ca)^\N$, ${\rm Aut}(\Q,<)$, $\mathrm{Homeo}(\mathbb{R})$, or $\mathrm{Homeo}(S^1)$ 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 on V.G. Pestov, that any action of the discrete group ${\mathrm{Homeo}}_{+}(\mathbb{R})$ by homeomorphisms on a compact metric space has a fixed point.