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Isolated circular orders on free products of cyclic groups

Chihaya Jibiki, Shuhei Maruyama

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.

Isolated circular orders on free products of cyclic groups

Abstract

In this paper, we construct countably many isolated circular orders on the free products 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 -extension of , which are not the automorphic images of the others.

Paper Structure

This paper contains 13 sections, 33 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Circular orders
  4. Left orders
  5. Known results on isolated circular orders on free products of cyclic groups
  6. Isolated circular orders
  7. Construction of circular orders
  8. Free subgroup of $G$
  9. Isolation of $c$
  10. Infinitely many isolated orders on $G$
  11. Isolated left orders
  12. Isolated left orders on a central $\mathbb{Z}$-extension
  13. Proof of Theorem \ref{['thm:main_left']}

Key Result

Theorem 1.1

Let $n \in \mathbb{Z}_{\geq 0}, k \in \mathbb{Z}_{\geq 1}$ and $m_1, \ldots, m_k \in \mathbb{Z}_{\geq 2}$ such that $(n,k,m_1, \cdots, m_k) \neq (0,1,m_1), (0,2,2,2)$. Then the free product $G = F_{2n} \ast \mathbb{Z}_{m_1} \ast \cdots \ast \mathbb{Z}_{m_k}$ admits countably many isolated circular o

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: MR3887426
  • Theorem 2.4: MR3887426
  • Theorem 2.5: MR3887426
  • Definition 2.6
  • Definition 2.7: MR3887426
  • Theorem 2.8: MR3887426
  • ...and 46 more