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ISSN (electronic): 1464-8997
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Circular groups, planar groups, and the Euler class

Danny Calegari

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 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.

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