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

摘要

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 F_(2) X 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 linear one.) Moreover, we show that for every non-solvable subgroup G of SL(2, Z), the group G X Z~(2) does not embed into Diff_(+)~(1)(S~(1)).