Geometry & Topology Monographs 7 (2004) 431–491

DOI: 10.2140/gtm.2004.7.431

Abstract

We study groups of C¹ 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^∞ diffeomorphisms of the plane is an unbounded class, and that any closed surface group of genus >1 admits a C^∞ action with arbitrary Euler class. On the other hand, we show that Z⊕Z actions satisfy a homological rigidity property: every orientation-preserving C¹ action of Z⊕Z on the plane has trivial Euler class. This gives the complete homological classification of surface group actions on R² in every degree of smoothness.

Keywords

Euler class, group actions, surface dynamics, braid groups, C¹ actions

Mathematical Subject Classification

Primary: 37C85

Secondary: 37E30, 57M60

Publication

Received: 9 September 2003
Revised: 30 July 2004
Accepted: 1 November 2004
Published: 13 December 2004

Authors