Low complexity among principal fully irreducible elements of Out($F_3$)

TL;DR

This paper determines the minimal stretch factor for principal fully irreducible elements in Out(F_3) and proves its uniqueness among single-fold constructions, advancing understanding of automorphisms of free groups.

Contribution

It identifies the shortest stretch factor for principal fully irreducible elements in Out(F_3) and establishes its uniqueness among single-fold elements.

Findings

Shortest realized stretch factor for Out(F_3) fully irreducible elements

Uniqueness of the principal element with this minimal stretch factor

Characterization of single-fold principal fully irreducible elements

Abstract

We find the shortest realized stretch factor for a fully irreducible $\varphi\in\mathrm{Out}(F_3)$ and show that it is realized by a "principal" fully irreducible element. We also show that it is the only principal fully irreducible produced by a single fold in any rank.