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