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

Some applications of H\"older's theorem in groups of analytic diffeomorphisms of 1-manifolds