Any Baumslag-Solitar action on surfaces with a pseudo-Anosov element has a finite orbit

Abstract

We consider $f, h$ homeomorphims generating a faithful $BS(1,n)$-action on a closed surface $S$, that is, $h f h^{-1} = f^n$, for some $ n\geq 2$. According to \cite{GL}, after replacing $f$ by a suitable iterate if necessary, we can assume that there exists a minimal set $\Lambda$ of the action, included in $Fix(f)$. Here, we suppose that $f$ and $h$ are $C^1$ in neighbourhood of $\Lambda$ and any point $x\in\Lambda$ admits an $h$-unstable manifold $W^u(x)$. Using Bonatti's techniques, we prove that either there exists an integer $N$ such that $W^u(x)$ is included in $Fix(f^N)$ or there is a lower bound for the norm of the differential of $h$ only depending on $n$ and the Riemannian metric on $S$. Combining last statement with a result of \cite{AGX}, we show that any faithful action of $BS(1, n)$ on $S$ with $h$ a pseudo-Anosov homeomorphism has a finite orbit. As a consequence, there is no faithful $C^1$-action of $BS(1, n)$ on the torus with $h$ an Anosov.