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 (.e., 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 semi-direct product between F_2 an Z^2, although locally indicable, does not embed into Diff_+^1 (]0,1[). (Here F_2 is any free subgroup of SL(2,Z), and its action on Z^2 is the projective one.) Moreover, we show that for every non-solvable subgroup G of SL(2,Z), the semi-direct product between G and Z^2 does not embed into Diff^1_+(S^1).

8 pages, no figure

References in corpus (4)

• Dense embeddings of surface groups
• Amenable groups that act on the line
• C^1 actions of the mapping class group on the circle
• Nonsmoothable, locally indicable group actions on the interval

Cited by in corpus (6)

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• A group with Property (T) acting on the circle
• On free discrete subgroups of Diff(I)