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Free products of bi-orderable groups

Kyrylo Muliarchyk

Abstract

We prove a bi-ordered version of Rivas' result for free products of left-order groups. Namely, we show that a free product of bi-ordered groups does not admit isolated bi-ordering. Our method relies on the dynamical realization of bi-ordered groups. We also show that the natural action of the automorphism group $Aut(F_2)$ on $F_2$ does not have dense orbits.

Free products of bi-orderable groups

Abstract

We prove a bi-ordered version of Rivas' result for free products of left-order groups. Namely, we show that a free product of bi-ordered groups does not admit isolated bi-ordering. Our method relies on the dynamical realization of bi-ordered groups. We also show that the natural action of the automorphism group on does not have dense orbits.
Paper Structure (4 sections, 10 theorems, 42 equations)

This paper contains 4 sections, 10 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Dynamical Realization
  3. Free Products
  4. Absence of dense orbits

Key Result

Theorem 1

Let $G$ and $H$ be two bi-orderable groups. Then the free product $G\bigast H$ has no isolated bi-orderings.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Remark 1
  • proof : Proof of Theorem \ref{['dynre']}
  • Proposition 1
  • proof
  • ...and 7 more