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

TL;DR

This paper constructs examples of finitely generated, locally indicable groups that cannot be faithfully represented by C^1 diffeomorphisms of the interval or circle, disproving the converse of Thurston's stability theorem.

Contribution

It provides explicit counterexamples of locally indicable groups that do not embed into Diff^1, showing the converse of Thurston's theorem is false.

Findings

Semi-direct product of F_2 and Z^2 does not embed into Diff_+^1(0,1).

Semi-direct product of non-solvable G in SL(2,Z) and Z^2 does not embed into Diff^1_+(S^1).

Counterexamples to the converse of Thurston's stability theorem.

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).