Order-Preserving outer automorphisms of free and surface groups

TL;DR

This paper classifies when finite outer automorphism groups preserve bi-orders on free and surface groups, introduces a new criterion related to braid actions, and provides examples answering a prior open question.

Contribution

It offers a complete classification of order-preserving outer automorphisms and introduces a novel criterion linking braid actions to bi-order preservation.

Findings

Classified when finite outer automorphism groups preserve bi-orders on free and surface groups.

Developed a new criterion for automorphisms induced by braid actions to preserve bi-orders.

Produced examples with full cycle permutations that affirm an open question.

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.