Automatic continuity of homomorphisms and fixed points on metric compacta

Details Automatic continuity of homomorphisms and fixed points on metric compacta Automatic continuity of homomorphisms and fixed points on metric compacta

2006-04-26

arxiv.org/abs/math/0604575v1

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Logic

Scientific paper

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

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