Spaces of invariant circular orders of groups

TL;DR

This paper introduces a topology on the space of all left-invariant circular orders on groups, explores its properties, and applies it to classify and analyze specific classes of groups, linking order theory with topology and dynamics.

Contribution

It defines a new topology on CO(G), studies its properties, and applies it to classify order spaces and analyze group actions, extending previous work on linear orders.

Findings

The space CO(G) is compact and contains LO(G) as a closed subspace.

The set of non-orderable finitely presented groups is recursively enumerable.

Aut(G) acts faithfully on certain subspaces of CO(G).

Abstract

Motivated by well known results in low-dimensional topology, we introduce and study a topology on the set CO(G) of all left-invariant circular orders on a fixed countable and discrete group G. CO(G) contains as a closed subspace LO(G), the space of all left-invariant linear orders of G, as first topologized by Sikora. We use the compactness of these spaces to show the sets of non-linearly and non-circularly orderable finitely presented groups are recursively enumerable. We describe the action of Aut(G) on CO(G) and relate it to results of Koberda regarding the action on LO(G). We then study two families of circularly orderable groups: finitely generated abelian groups, and free products of circularly orderable groups. For finitely generated abelian groups A, we use a classification of elements of CO(A) to describe the homeomorphism type of the space CO(A), and to show that Aut(A) acts faithfully on the subspace of circular orders which are not linear. We define and characterize Archimedean circular orders, in analogy with linear Archimedean orders. We describe explicit examples of circular orders on free products of circularly orderable groups, and prove a result about the abundance of orders on free products. Whenever possible, we prove and interpret our results from a dynamical perspective.