Title: 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.