Free planar actions of the Klein bottle group

TL;DR

This paper characterizes the structure of free, orientation-preserving actions of the Klein bottle group on the plane, revealing that one generator acts properly discontinuously while the other cannot, and explores implications for torsion-free groups.

Contribution

It provides a detailed description of free planar actions of the Klein bottle group and identifies constraints on the actions of its generators, advancing understanding of group actions on the plane.

Findings

$a$ acts properly discontinuously

$b$ cannot act properly discontinuously

Some torsion-free groups cannot act freely on the plane

Abstract

We describe the structure of the free actions of the Klein bottle group by orientation preserving homeomorphisms of the plane. This group is generated by two elements $a,b$, where the conjugate of $b$ by $a$ equals the inverse of $b$. The main result is that $a$ must act properly discontinuously, while $b$ cannot act properly discontinuously. As a corollary, we describe some torsion free groups that cannot act freely on the plane. We also find some properties which are reminiscent of Brouwer theory for the infinite cyclic group $Z$, in particular that every free action is virtually wandering.