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According to Thurston's stability theorem, every group of C^1 diffeomorphismsof the closed interval is locally indicable (.e., every finitely generatedsubgroup factors through Z). We show that, even for finitely generated groups,the converse of this statement is not true. More precisely, we show that thesemi-direct product between F_2 an Z^2, although locally indicable, does notembed into Diff_+^1 (]0,1[). (Here F_2 is any free subgroup of SL(2,Z), and itsaction on Z^2 is the projective one.) Moreover, we show that for everynon-solvable subgroup G of SL(2,Z), the semi-direct product between G and Z^2does not embed into Diff^1_+(S^1).

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