Planar Baumslag-Solitar actions

TL;DR

This paper investigates specific types of group actions of Baumslag-Solitar groups on planar and toral homeomorphisms, focusing on elliptic, parabolic, and linear cases, and establishes non-existence results for certain faithful representations.

Contribution

It extends previous work by analyzing elliptic and parabolic cases and proves the non-existence of faithful toral representations with Dehn twist conjugation.

Findings

No faithful representations of BS(1,n) by toral homeomorphisms with Dehn twist as conjugating element.

Extended analysis of elliptic and parabolic cases in planar actions.

Clarified limitations of Baumslag-Solitar group actions on surfaces.

Abstract

Following previous work, where representations of $BS(1,n)$ by planar orientation preserving homeomorphisms with linear diagonalizable conjugating element was studied, we consider the elliptic and parabolic cases. As an application, we prove that there are no faithful representations of $BS(1,n)$ by toral homeomorphisms with conjugating element the Dehn twist map.