#### 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

Email: dannyc@its.caltech.edu

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