Volume 7 (2004)

Download This Article
with up-to-date links in citations
For Screen
For Printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
All Volumes
About this Series
Ethics Statement
Purchase Printed Copies
Author Index
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
MSP Books and Monographs
Other MSP Publications

Circular groups, planar groups, and the Euler class

Danny Calegari

Geometry & Topology Monographs 7 (2004) 431–491

DOI: 10.2140/gtm.2004.7.431


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 ZZ actions satisfy a homological rigidity property: every orientation-preserving C1 action of ZZ on the plane has trivial Euler class. This gives the complete homological classification of surface group actions on R2 in every degree of smoothness.


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

Mathematical Subject Classification

Primary: 37C85

Secondary: 37E30, 57M60


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

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