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A new characterization of Conrad's property for group orderings, with applications

Andrés Navas and Cristóbal Rivas

Appendix: Adam Clay

Algebraic & Geometric Topology 9 (2009) 2079–2100

We provide a pure algebraic version of the first-named author’s dynamical characterization of the Conrad property for group orderings. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof of a theorem first established by Linnell: an orderable group having infinitely many orderings has uncountably many. This proof is achieved by extending to uncountable orderable groups a result about orderings which may be approximated by their conjugates. This last result is illustrated by an example of an exotic ordering on the free group given by the third author in the Appendix.

group orders, Conrad's property
Mathematical Subject Classification 2000
Primary: 06F15, 20F60
Secondary: 57S25
Received: 3 March 2009
Revised: 28 August 2009
Accepted: 31 August 2009
Published: 16 October 2009
Andrés Navas
Departamento de Matemática y Ciencia de la Computación
Facultad de Ciencia
Universidad de Santiago de Chile
Alameda 3363
Estación Central
Cristóbal Rivas
Departamento de Matemáticas
Facultad de Ciencia
Universidad de Chile
Las Palmeras 3425
Adam Clay
Department of Mathematics
University of British Columbia
British Columbia
V6T 1Z2