## Geometry & Topology

### Free planar actions of the Klein bottle group

Frédéric Le Roux

#### Abstract

We describe the structure of the free actions of the fundamental group of the Klein bottle $〈a,b∣aba−1=b−1〉$ by orientation preserving homeomorphisms of the plane. 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 may not act freely on the plane. We also find some properties which are reminiscent of Brouwer theory for the group $ℤ$, in particular that every free action is virtually wandering.

#### Article information

Source
Geom. Topol., Volume 15, Number 3 (2011), 1545-1567.

Dates
Revised: 25 January 2011
Accepted: 29 June 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732339

Digital Object Identifier
doi:10.2140/gt.2011.15.1545

Mathematical Reviews number (MathSciNet)
MR2851071

Zentralblatt MATH identifier
1268.37066

#### Citation

Le Roux, Frédéric. Free planar actions of the Klein bottle group. Geom. Topol. 15 (2011), no. 3, 1545--1567. doi:10.2140/gt.2011.15.1545. https://projecteuclid.org/euclid.gt/1513732339

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