Some applications of H\"older'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 $\mathrm{Diff}_{+}^{\omega }(\mathbb{S}^1)$. We also prove that a branch group does not embed into $\mathrm{Diff}_{+}^{\omega }(\mathbb{S}^1)$. 
Azer Akhmedov and Michael P. Cohen
math.GR math.DS