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$C^1$–actions of Baumslag–Solitar groups on $S^1$

Nancy Guelman and Isabelle Liousse

Algebraic & Geometric Topology 11 (2011) 1701–1707

Let BS(1,n) = a,baba1 = bn be the solvable Baumslag–Solitar group, where n 2. It is known that BS(1,n) is isomorphic to the group generated by the two affine maps of the line: f0(x) = x + 1 and h0(x) = nx. The action on S1 = generated by these two affine maps f0 and h0 is called the standard affine one. We prove that any faithful representation of BS(1,n) into Diff1(S1) is semiconjugated (up to a finite index subgroup) to the standard affine action.

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circle diffeomorphism, solvable Baumslag–Solitar group
Mathematical Subject Classification 2010
Primary: 37C85
Secondary: 57S25, 37E10
Received: 28 October 2010
Revised: 6 April 2011
Accepted: 9 April 2011
Published: 3 June 2011
Nancy Guelman
Facultad de Ingeniería
Universidad de la República
Julio Herrera y Reissig 565
11300 Montevideo
Isabelle Liousse
UFR de Mathématiques
UMR CNRS 8524, Université de Lille 1
Laboratoire Paul Painlevé
59655 Villeneuve d’Ascq