Discretely ordered groups

Details Discretely ordered groups Discretely ordered groups

2008-08-20

arxiv.org/abs/0808.2686v2

Mathematics

Group Theory

9 pages, minor changes, to appear in Algebra and Number theory

Scientific paper

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 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 non-locally-indicable right orderable groups.

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