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Order-Preserving outer automorphisms of free and surface groups

Jonathan Johnson, Khanh Le

TL;DR

This work classifies finite-order outer automorphisms that preserve bi-orders on non-abelian free groups and bi-orderable surface groups, showing that a finite subgroup $H$ of $ ext{Out}(G)$ is order-preserving iff its preimage in $ ext{Aut}(G)$ is torsion-free. It then introduces a Burau-representation based criterion for order-preservation of braid-induced outer automorphisms: if the reduced Burau representation $ ho(eta)$ has all eigenvalues positive in the real-closed Puiseux field $\mathbb{E}$, then $eta$ is order-preserving. This yields new order-preserving braids, including many with a single cycle permutation, and provides a protocol to generate infinite families of order-preserving examples. The results link bi-orderability with geometric realizability and offer algorithmic decision tools for finite-order cases, significantly advancing understanding of orderability in free and surface groups and their braid-induced automorphisms.

Abstract

We give a complete classification to when a finite group of outer automorphisms preserves a bi-order on a non-abelian free group and bi-orderable surface groups. We also give another new criterion for an outer automorphism of $F_n$ induced by action of an $n$-strand braid to preserve a bi-order on $F_n.$ Using the new criterion, we produce examples of order-preserving whose underlying permutation is a full cycle which answers in affirmative a question of Kin and Rolfsen.

Order-Preserving outer automorphisms of free and surface groups

TL;DR

This work classifies finite-order outer automorphisms that preserve bi-orders on non-abelian free groups and bi-orderable surface groups, showing that a finite subgroup of is order-preserving iff its preimage in is torsion-free. It then introduces a Burau-representation based criterion for order-preservation of braid-induced outer automorphisms: if the reduced Burau representation has all eigenvalues positive in the real-closed Puiseux field , then is order-preserving. This yields new order-preserving braids, including many with a single cycle permutation, and provides a protocol to generate infinite families of order-preserving examples. The results link bi-orderability with geometric realizability and offer algorithmic decision tools for finite-order cases, significantly advancing understanding of orderability in free and surface groups and their braid-induced automorphisms.

Abstract

We give a complete classification to when a finite group of outer automorphisms preserves a bi-order on a non-abelian free group and bi-orderable surface groups. We also give another new criterion for an outer automorphism of induced by action of an -strand braid to preserve a bi-order on Using the new criterion, we produce examples of order-preserving whose underlying permutation is a full cycle which answers in affirmative a question of Kin and Rolfsen.

Paper Structure

This paper contains 17 sections, 27 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Organization of the article
  3. Acknowledgement
  4. Classification of finite-order outer automorphisms of free and surface groups
  5. Motivating examples
  6. The braid $\sigma_1^2$ in $B_3$ is order-preserving.
  7. Remarks about the example and the generalization
  8. Preliminaries
  9. Tensor product construction and properties
  10. Ordered rings and fields
  11. The field of Puiseux series and its ordering
  12. Braids and the Burau representation
  13. Order-preserving braid via Burau representation
  14. Proof of the main theorem
  15. Applications to order-preserving 3-braids
  16. ...and 2 more sections

Key Result

Proposition 1.1

Let $G$ be an extension of two bi-orderable groups $K$ and $Q$. In particular, we have the following short exact sequence of groups where $K$ and $Q$ are bi-orderable groups. Let $\rho:Q \to \mathop{\mathrm{Out}}\nolimits(K)$ be the action associated to the short exact sequence. If there exists a bi-order $P \in \mathop{\mathrm{BO}}\nolimits(K)$ such that $\rho(Q)$ fixes $P$, then $G$ is bi-order

Theorems & Definitions (34)

  • Proposition 1.1
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5: PerronRolfsen03
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 2.1
  • Definition 4.1
  • ...and 24 more