Geometry & Topology

Free planar actions of the Klein bottle group

Frédéric Le Roux

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We describe the structure of the free actions of the fundamental group of the Klein bottle a,baba1=b1 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 , in particular that every free action is virtually wandering.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 57S25: Groups acting on specific manifolds

plane homeomorphism free group action


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.

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