Order-Preserving outer automorphisms of free and surface groups

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.

25 pages. Revised according to comments from the referees. The exposition in Section 2 was largely rewritten. Section 4.1 was removed. More details added to section 5.1. To appear in Journal of Algebra