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

Abstract

Let $BS\left(1,n\right)=\langle a,b\mid ab{a}^{-1}=b^n\rangle$ 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.