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Ordered bases, order-preserving automorphisms and bi-orderable link groups

Tommy Wuxing Cai, Adam Clay, Dale Rolfsen

TL;DR

The paper develops a criterion to guarantee that a free group automorphism is order-preserving with a bi-ordering invariant, based on a coprimality condition on finite orbits of the induced permutation and a height map on generators. It lifts this criterion to arbitrary ranks via positively triangular abelianizations and tensor-compatibility, yielding a general method to construct bi-orderings invariant under free-group automorphisms. This framework is then applied to the Artin action of braids, showing that many braid-induced automorphisms preserve bi-orderings, and in particular proving that the fundamental group of the magic manifold is bi-orderable. The authors further provide a mechanism to enlarge any link to a bi-orderable one by adding two components and to turn non-order-preserving braids into order-preserving ones by multiplying by certain braids, with broader implications for Dehn surgery and 3-manifold topology. Collectively, these results give new tools for identifying bi-orderable link groups and expanding the class of known bi-orderable 3-manifolds.

Abstract

We give a new criterion which guarantees that a free group admits a bi-ordering that is invariant under a given automorphism. As an application, we show that the fundamental group of the "magic manifold" is bi-orderable, answering a question of Kin and Rolfsen.

Ordered bases, order-preserving automorphisms and bi-orderable link groups

TL;DR

The paper develops a criterion to guarantee that a free group automorphism is order-preserving with a bi-ordering invariant, based on a coprimality condition on finite orbits of the induced permutation and a height map on generators. It lifts this criterion to arbitrary ranks via positively triangular abelianizations and tensor-compatibility, yielding a general method to construct bi-orderings invariant under free-group automorphisms. This framework is then applied to the Artin action of braids, showing that many braid-induced automorphisms preserve bi-orderings, and in particular proving that the fundamental group of the magic manifold is bi-orderable. The authors further provide a mechanism to enlarge any link to a bi-orderable one by adding two components and to turn non-order-preserving braids into order-preserving ones by multiplying by certain braids, with broader implications for Dehn surgery and 3-manifold topology. Collectively, these results give new tools for identifying bi-orderable link groups and expanding the class of known bi-orderable 3-manifolds.

Abstract

We give a new criterion which guarantees that a free group admits a bi-ordering that is invariant under a given automorphism. As an application, we show that the fundamental group of the "magic manifold" is bi-orderable, answering a question of Kin and Rolfsen.

Paper Structure

This paper contains 9 sections, 16 theorems, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Acknowledgments
  4. Ordered bases, modules, and order-preserving homomorphisms
  5. Order-preserving automorphisms of free groups
  6. Order-preserving braids, bi-orderable link groups and 3-manifolds
  7. Minimal volume 3-manifolds
  8. Dehn surgery and sublinks of bi-orderable links
  9. Multiplication by braids in B(n-1)

Key Result

Lemma 2.4

Let $M$ be a free $R$-module and $\mathcal{B} = (\{v_i \mid i\in I\}, \prec)$ an ordered basis for $M$. If $f:M\rightarrow M$ is an $R$-module homomorphism that is positively triangular with respect to $\mathcal{B}$, then $f$ preserves the bi-ordering $<_{\mathcal{B}}$.

Figures (2)

  • Figure 1: The link $br(\beta)$ for $\beta = \sigma_1^2\sigma_2^{-1}$, whose complement is homeomorphic to the magic manifold.
  • Figure 2: The braid $\beta = \sigma_1 \sigma_3$ in the dashed box on top, and the braid $\alpha = A_{4,5} A_{2,5}$ in the dashed box on the bottom.

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 3.1
  • proof
  • ...and 29 more