#### Volume 14, issue 1 (2010)

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

### Andrés Navas

Geometry & Topology 14 (2010) 573–584
##### 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 $ℤ$). We show that, even for finitely generated groups, the converse of this statement is not true. More precisely, we show that the group ${\mathbb{F}}_{2}⋉{ℤ}^{2}$, although locally indicable, does not embed into ${Diff}_{+}^{1}\left(\left(0,1\right)\right)$. (Here ${\mathbb{F}}_{2}$ is any free subgroup of $SL\left(2,ℤ\right)$, and its action on ${ℤ}^{2}$ is the linear one.) Moreover, we show that for every non-solvable subgroup $G$ of $SL\left(2,ℤ\right)$, the group $G⋉{ℤ}^{2}$ does not embed into ${Diff}_{+}^{1}\left({S}^{1}\right)$.

##### Keywords
Thurston's stability, locally indicable group
##### Mathematical Subject Classification 2000
Primary: 20B27, 37C85, 37E05