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Circular groups, planar groups, and the Euler class

Calegari, Danny (2004) Circular groups, planar groups, and the Euler class. In: Proceedings of the Casson Fest (Arkansas and Texas 2003). Geometry & Topology Monographs. No.7. Geometry & Topology Publications , Coventry, UK, pp. 431-491.

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We study groups of C^1 orientation-preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly-orderable groups. We show that every such group with a bounded orbit is circularly-orderable, and show that certain generalized braid groups are circularly-orderable. We also show that the Euler class of C^infty diffeomorphisms of the plane is an unbounded class, and that any closed surface group of genus >1 admits a C^infty action with arbitrary Euler class. On the other hand, we show that Z oplus Z actions satisfy a homological rigidity property: every orientation-preserving C^1 action of Z oplus Z on the plane has trivial Euler class. This gives the complete homological classification of surface group actions on R^2 in every degree of smoothness.

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Additional Information:© 2004 Geometry & Topology Publications. Submitted to GT on 9 September 2003. (Revised 30 July 2004.) Paper accepted 1 November 2004. Paper published 13 December 2004. I would like to thank Mladen Bestvina, Nathan Dunfield, Bob Edwards, Benson Farb, John Franks, Étienne Ghys, Michael Handel, Dale Rolfsen, Frédéric le Roux, Takashi Tsuboi, Amie Wilkinson and the anonymous referee for some very useful conversations and comments. I would especially like to single out Étienne Ghys for thanks, for reading an earlier version of this paper and providing me with copious comments, observations, references, and counterexamples to some naive conjectures. While writing this paper, I received partial support from the Sloan foundation, and from NSF grant DMS-0405491.
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Alfred P. Sloan FoundationUNSPECIFIED
Subject Keywords:Euler class, group actions, surface dynamics, braid groups, C^1 actions
Series Name:Geometry & Topology Monographs
Issue or Number:7
Record Number:CaltechAUTHORS:CALgtm04
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1141
Deposited By: Archive Administrator
Deposited On:22 Dec 2005
Last Modified:08 Nov 2021 19:07

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