Calegari, Danny (2004) Circular groups, planar groups, and the Euler class. In: Proceedings of the Casson Fest (Arkansas and Texas 2003). Geometry & Topology Monographs (7). Geometry & Topology Publications , Coventry, UK, pp. 431-491. http://resolver.caltech.edu/CaltechAUTHORS:CALgtm04
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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|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Archive Administrator|
|Deposited On:||22 Dec 2005|
|Last Modified:||26 Dec 2012 08:42|
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