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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.

Item Type:Book Section
Additional Information: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, ´Etienne Ghys, Michael Handel, Dale Rolfsen, Fr´ed´eric le Roux, Takashi Tsuboi, Amie Wilkinson and the anonymous referee for some very useful conversations and comments. I would especially like to single out ´Etienne 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.
Subject Keywords:Euler class, group actions, surface dynamics, braid groups, C^1 actions
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:26 Dec 2012 08:42

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