One some planar Baumslag-Solitar actions

Abstract

Let $BS(1,n)=⟨a,b:ab{a}^{-1}=b^n⟩$ be the solvable Baumslag-Solitar group for $n\ge 2$. We study representations of $BS(1,n)$ 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 $BS(1,n)$ by homeomorphisms of the torus.

28 pages, 2 figures

References in corpus (1)

• Groups acting on manifolds: around the Zimmer program