Geometry & Topology Monographs, Vol. 7 (2004),
Proceedings of the Casson Fest,
Paper no. 15, pages 431--491.

Circular groups, planar groups, and the Euler class

Danny Calegari

Abstract. 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.

Keywords. Euler class, group actions, surface dynamics, braid groups, C^1 actions

AMS subject classification. Primary: 37C85. Secondary: 37E30, 57M60.

E-print: arXiv:math.GT/0403311

Submitted to GT on 9 September 2003. (Revised 30 July 2004.) Paper accepted 1 November 2004. Paper published 13 December 2004.

Notes on file formats

Danny Calegari
Department of Mathematics, California Institute of Technology
Pasadena CA, 91125, USA

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