On some planar Baumslag–Solitar actions

Abstract

Let $\mathrm{BS}(1,n)=⟨a,b:ab{a}^{-1}=b^n⟩$ be the solvable Baumslag–Solitar group for $n\ge 2$. We study representations of $\mathrm{BS}(1,n)$ on the plane by orientation-preserving homeomorphisms, assuming that $a$ acts as a linear map and $b$ as a map with bounded displacement. We find that the possibilities for a faithful action depend greatly on the Jordan canonical form of the map $h$ defined by the action of $a$. In case $h$ is diagonalizable over $\mathbb{R}$, we shall give examples or prove rigidity theorems depending on the eigenvalues. We also show some rigidity in the cases where $h$ is elliptic or parabolic. Then we give applications to the actions of $\mathrm{BS}(1,n)$ by homeomorphisms of the torus.