Discretely ordered groups

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.