#### Volume 15, issue 3 (2011)

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Free planar actions of the Klein bottle group

### Frédéric Le Roux

Geometry & Topology 15 (2011) 1545–1567
##### Abstract

We describe the structure of the free actions of the fundamental group of the Klein bottle $〈a,b\mid ab{a}^{-1}={b}^{-1}〉$ 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.

##### Keywords
plane homeomorphism, free group action
##### Mathematical Subject Classification 2000
Primary: 37E30, 57S25