Volume 11, issue 3 (2011)

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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
Abstract

Let $BS\left(1,n\right)=〈a,b\mid ab{a}^{-1}={b}^{n}〉$ be the solvable Baumslag–Solitar group, where $n\ge 2$. It is known that $BS\left(1,n\right)$ is isomorphic to the group generated by the two affine maps of the line: ${f}_{0}\left(x\right)=x+1$ and ${h}_{0}\left(x\right)=nx$. The action on ${S}^{1}=ℝ\cup \infty$ generated by these two affine maps ${f}_{0}$ and ${h}_{0}$ is called the standard affine one. We prove that any faithful representation of $BS\left(1,n\right)$ into ${Diff}^{1}\left({S}^{1}\right)$ is semiconjugated (up to a finite index subgroup) to the standard affine action.

Keywords
circle diffeomorphism, solvable Baumslag–Solitar group
Mathematical Subject Classification 2010
Primary: 37C85
Secondary: 57S25, 37E10