Discretely ordered groups

Linnell, Peter; Rhemtulla, Akbar; Rolfsen, Dale

doi:10.2140/ant.2009.3.797

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Discretely ordered groups

Peter A Linnell, Akbar Rhemtulla and Dale P. O. Rolfsen

Vol. 3 (2009), No. 7, 797–807

DOI: 10.2140/ant.2009.3.797

Abstract

We consider group orders and right-orders which are discrete, meaning there is a least element which is greater than the identity. We note that nonabelian free groups cannot be given discrete orders, although they do have right-orders which are discrete. More generally, we give necessary and sufficient conditions that a given orderable group can be endowed with a discrete order. In particular, every orderable group $G$ embeds in a discretely orderable group. We also consider conditions on right-orderable groups to be discretely right-orderable. Finally, we discuss a number of illustrative examples involving discrete orderability, including the Artin braid groups and Bergman’s nonlocally-indicable right orderable groups.

Keywords

discrete order

Mathematical Subject Classification 2000

Primary: 20F60

Secondary: 06F15, 20F36

Milestones

Received: 19 August 2008

Revised: 9 March 2009

Accepted: 17 April 2009

Published: 29 November 2009

Authors

Peter A Linnell
Department of Mathematics Virginia Tech Blacksburg, VA 24061-0123 United States
http://www.math.vt.edu/people/plinnell/
Akbar Rhemtulla
Department of Mathematical & Statistical Sciences University of Alberta Edmonton, AL T6G 2G1 Canada
http://www.math.ualberta.ca/Rhemtulla_A.html
Dale P. O. Rolfsen
University of British Columbia Mathematics Department 1984 Mathematics Road Vancouver, BC V6T 1Z2 Canada
http://www.math.ubc.ca/~rolfsen/

Vol. 3, No. 7, 2009