Algebraic & Geometric Topology

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

Nancy Guelman and Isabelle Liousse

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Let BS(1,n)=a,baba1=bn be the solvable Baumslag–Solitar group, where n2. 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.

Article information

Algebr. Geom. Topol., Volume 11, Number 3 (2011), 1701-1707.

Received: 28 October 2010
Revised: 6 April 2011
Accepted: 9 April 2011
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx]
Secondary: 57S25: Groups acting on specific manifolds 37E10: Maps of the circle

circle diffeomorphism solvable Baumslag–Solitar group


Guelman, Nancy; Liousse, Isabelle. $C^1$–actions of Baumslag–Solitar groups on $S^1$. Algebr. Geom. Topol. 11 (2011), no. 3, 1701--1707. doi:10.2140/agt.2011.11.1701.

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