One some planar Baumslag-Solitar actions

TL;DR

This paper investigates how the Baumslag-Solitar group BS(1,n) can act on the plane via orientation-preserving homeomorphisms, revealing that the nature of the linear part of the action critically influences the possibility of faithful representations.

Contribution

It classifies and analyzes the possible faithful actions of BS(1,n) on the plane based on the Jordan form of the linear map, providing new rigidity results and examples.

Findings

Faithful actions depend on the Jordan form of the linear map h.

Rigidity results for diagonalizable, elliptic, and parabolic cases.

Applications to torus homeomorphisms.

Abstract

Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvable Baumslag-Solitar group for $n \geq 2$. 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.