Circular groups, planar groups, and the Euler class
Abstract
We study groups of C^1 orientationpreserving homeomorphisms of the plane, and pursue analogies between such groups and circularlyorderable groups. We show that every such group with a bounded orbit is circularlyorderable, and show that certain generalized braid groups are circularlyorderable. 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 orientationpreserving 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2004
 arXiv:
 arXiv:math/0403311
 Bibcode:
 2004math......3311C
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Dynamical Systems;
 Mathematics  Group Theory;
 37C85;
 37E30;
 57M60
 EPrint:
 Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon7/paper15.abs.html