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Let $G$ be a free group, if the action of $G$ on $\mathbb{R}$ is free, does it imply that $G$ is abelian?

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I think this should come down mostly to definitions. –  Newb Jun 13 at 14:25
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"the" action of $G$ on $\mathbb R$? Is there supposed to be a fixed action here? Are there any restrictions on this action in terms of the structure of $G$ and $\mathbb R$? –  Dustan Levenstein Jun 13 at 14:43

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Hölder has proved that any group $G$ which acts freely on the real line by homeomorphisms is abelian.

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