Vol. 3, No. 7, 2009

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Discretely ordered groups

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

Vol. 3 (2009), No. 7, 797–807

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.

discrete order
Mathematical Subject Classification 2000
Primary: 20F60
Secondary: 06F15, 20F36
Received: 19 August 2008
Revised: 9 March 2009
Accepted: 17 April 2009
Published: 29 November 2009
Peter A Linnell
Department of Mathematics
Virginia Tech
Blacksburg, VA 24061-0123
United States
Akbar Rhemtulla
Department of Mathematical & Statistical Sciences
University of Alberta
Edmonton, AL  T6G 2G1
Dale P. O. Rolfsen
University of British Columbia
Mathematics Department
1984 Mathematics Road
Vancouver, BC  V6T 1Z2