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

Abstract

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 ${\mathrm{Diff}}_{+}^{\omega }({\mathbb{S}}^1)$.