The space of circular orderings and semiconjugacy

TL;DR

This paper generalizes Linnell's dichotomy on the size of the space of left-orderings of a group by introducing semiconjugacy of circular orderings, showing each semiconjugacy class is either finite or uncountably infinite, with structural implications.

Contribution

It introduces a new framework of semiconjugacy for circular orderings, extending Linnell's results to a broader setting and analyzing the structure of orderings within this framework.

Findings

Semiconjugacy classes of circular orderings are either finite or uncountably infinite.

Finite semiconjugacy classes correspond to groups with specific structural properties.

The space of left-orderings embeds as a subspace within the space of circular orderings.

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.