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

Abstract

We consider homeomorphisms $f,h$ generating a faithful $\mathit{BS}(1,n)$-action on a closed surface $S$, that is, $hfh^{-1}=f^{n}$ for some $n\geq 2$. According to Guelman and Liousse [Actions of Baumslag–Solitar groups on surfaces. Discrete Contin. Dyn. Syst. A 5 (2013), 1945–1964], after replacing $f$ by a suitable iterate if necessary, we can assume that there exists a minimal set $\unicode[STIX]{x1D6EC}$ of the action, included in $\text{Fix}(f)$. Here, we suppose that $f$ and $h$ are $C^{1}$ in a neighborhood of $\unicode[STIX]{x1D6EC}$ and any point $x\in \unicode[STIX]{x1D6EC}$ 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 $\text{Fix}(f^{N})$ or there is a lower bound for the norm of the differential of $h$ depending only on $n$ and the Riemannian metric on $S$. Combining the last statement with a result of Alonso, Guelman and Xavier [Actions of solvable Baumslag–Solitar groups on surfaces with (pseudo)-Anosov elements. Discrete Contin. Dyn. Syst. to appear], we show that any faithful action of $\mathit{BS}(1,n)$ on $S$ with $h$ a pseudo-Anosov homeomorphism has a finite orbit containing singularities of $h$; moreover, if $f$ is isotopic to the identity, it is entirely contained in the singular set of $h$. As a consequence, there is no faithful $C^{1}$-action of $\mathit{BS}(1,n)$ on the torus with $h$ Anosov.