Isolated circular orders on free products of cyclic groups

TL;DR

This paper constructs numerous distinct isolated circular orders on free products of cyclic groups and extends these to isolated left orders on certain central extensions, revealing new structural properties.

Contribution

It introduces a method to construct countably many isolated circular orders on free products of cyclic groups and their central extensions, showing they are not automorphic images.

Findings

Countably many isolated circular orders constructed

These orders are not automorphic images of each other

Constructs isolated left orders on central extensions

Abstract

In this paper, we construct countably many isolated circular orders on the free products $G = F_{2n} \ast \mathbb{Z}_{m_1} \ast \cdots \ast \mathbb{Z}_{m_k}$ of cyclic groups. Moreover, we prove that these isolated circular orders are not the automorphic images of the others. By using these isolated circular orders, we also construct countably many isolated left orders on a certain central $\mathbb{Z}$-extension of $G$, which are not the automorphic images of the others.