A finitely generated, locally indicable group with no faithful action by $C^1$ diffeomorphisms of the interval

Abstract

According to Thurston’s stability theorem, every group of $C^1$ diffeomorphisms of the closed interval is locally indicable (that is, every finitely generated subgroup factors through $\mathbb{Z}$). We show that, even for finitely generated groups, the converse of this statement is not true. More precisely, we show that the group ${\mathbb{F}}_2⋉{\mathbb{Z}}^2$, although locally indicable, does not embed into ${Diff}_{+}^1\left(\left(0,1\right)\right)$. (Here ${\mathbb{F}}_2$ is any free subgroup of $SL\left(2,\mathbb{Z}\right)$, and its action on ${\mathbb{Z}}^2$ is the linear one.) Moreover, we show that for every non-solvable subgroup $G$ of $SL\left(2,\mathbb{Z}\right)$, the group $G⋉{\mathbb{Z}}^2$ does not embed into ${Diff}_{+}^1\left(S^1\right)$.