# On the number of circular orders on a group

## TL;DR

This paper classifies groups with finitely many circular orders, showing they are solvable with specific structures, and reveals the nature of the space of circular orders as either finite or uncountably infinite.

## Contribution

It provides a complete algebraic classification of groups with finitely many circular orders and describes the topological structure of the space of all circular orders.

## Key findings

- Groups with finitely many circular orders are solvable with a specific semi-direct product structure.
- The space of circular orders is either finite or uncountably infinite.
- The space of circular orders on an infinite Abelian group is a Cantor set.

## Abstract

We give a classification and complete algebraic description of groups allowing only finitely many (left multiplication invariant) circular orders. In particular, they are all solvable groups with a specific semi-direct product decomposition. This allows us to also show that the space of circular orders of any group is either finite or uncountable. As a special case and first step, we show that the space of circular orderings of an infinite Abelian group has no isolated points, hence is homeomorphic to a cantor set.