Some applications of Hölder's theorem in groups of analytic diffeomorphisms of 1-manifolds

We obtain a simple obstruction to embedding groups into the analytic diffeomorphism groups of 1-manifolds. Using this, we classify all RAAGs which embed into $\mathrm{Diff}_{+}^{ω}(\mathbb{S}^1)$. We also prove that a branch group does not embed into $\mathrm{Diff}_{+}^{ω}(\mathbb{S}^1)$.

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