Free planar actions of the Klein bottle group

Le Roux, Frédéric

doi:10.2140/gt.2011.15.1545

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Free planar actions of the Klein bottle group

Frédéric Le Roux

Geometry & Topology 15 (2011) 1545–1567

DOI: 10.2140/gt.2011.15.1545

Abstract

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

References

Publication

Received: 25 January 2011

Revised: 25 January 2011

Accepted: 29 June 2011

Published: 5 September 2011

Proposed: Danny Calegari

Seconded: Leonid Polterovich, Walter Neumann

Authors

Frédéric Le Roux
Laboratoire de Mathématiques, CNRS UMR 8628 Université Paris Sud 11 F-91405 Orsay Cedex France
http://www.math.u-psud.fr/~leroux/

Volume 15, issue 3 (2011)