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

TL;DR

This paper uses H"older's theorem to identify obstructions to embedding certain groups into the analytic diffeomorphism groups of 1-manifolds, classifying RAAGs that embed and showing branch groups do not.

Contribution

It introduces a simple obstruction criterion for embeddings and classifies all RAAGs embeddable into the analytic diffeomorphisms of the circle.

Findings

Classified all RAAGs that embed into $ ext{Diff}_+^ ext{ω}( ext{S}^1)$

Proved that branch groups do not embed into $ ext{Diff}_+^ ext{ω}( ext{S}^1)$

Provided a new obstruction criterion based on H"older's theorem

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