Smooth actions of Baumslag-Solitar groups on the interval

arXiv:2611.13171v1 Announce Type: new Abstract: We study representations of the Baumslag-Solitar group $BS(m,n)=\langle a,b\mid ab^ma^{-1}=b^n\rangle$ inside the group of orientation-preserving diffeomorphisms of the interval $[0,1]$. We prove that for integers $n,m\in \mathbb Z\setminus\{-1,0,1\}$, the group $BS(m,n)$ embeds into $\mathrm{Diff}_+^1([0,1])$ if and only if $|n|=|m|$. Furthermore, we show that for any $\alpha\in (0,1)$, the group $BS(m,n)$ embeds inside $\mathrm{Diff}_+^{1+\alpha}([0,1])$ if and only if $1=|m|=|n|$ or when $1=m\leq n$.

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