The space of circular orderings and semiconjugacy

Abstract: Work of Linnell shows that the space of left-orderings of a group is either finite or uncountable, and in the case that the space is finite, the isomorphism type of the group is known---it is what is known as a Tararin group. By defining semiconjugacy of circular orderings in a general setting (that is, for arbitrary circular orderings of groups that may not act on $S^1$), we can view the subspace of left-orderings of any group as a single semiconjugacy class of circular orderings. Taking this perspective, we generalize the result of Linnell, to show that every semiconjugacy class of circular orderings is either finite or uncountable, and when a semiconjugacy class is finite, the group has a prescribed structure. We also investigate the space of left-orderings as a subspace of the space of circular orderings, addressing a question of Baik and Samperton.