Free planar actions of the Klein bottle group

Le Roux, Frédéric¹

¹ Laboratoire de Mathématiques, CNRS UMR 8628, Université Paris Sud 11, F-91405 Orsay Cedex, France

Geometry & topology, Tome 15 (2011) no. 3, pp. 1545-1567.

Résumé

We describe the structure of the free actions of the fundamental group of the Klein bottle $\langle a,b\mid ab{a}^{-1}=b^{-1}\rangle$ 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 $\mathbb{Z}$, in particular that every free action is virtually wandering.

Détail

DOI : 10.2140/gt.2011.15.1545

Keywords: plane homeomorphism, free group action

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