Fibered commensurability on $\mathrm{Out}(F_{n})$

TL;DR

This paper introduces fibered commensurability for outer automorphisms of free groups, extending concepts from mapping class groups, and proves the existence of unique minimal elements within each class under specific conditions.

Contribution

It defines fibered commensurability for outer automorphisms and establishes the existence of unique minimal elements in each class under certain invariance conditions.

Findings

Atoroidal and fully irreducible automorphisms are commensurability invariants.

Existence of unique minimal elements in each fibered commensurability class.

Conditions involving asymmetry of Whitehead graphs are crucial.

Abstract

We define and discuss a notion called fibered commensurability of outer automorphisms of free groups. This notion lets us study symmetry of outer automorphisms. The notion of fibered commensurability is first defined by Calegari-Sun-Wang on mapping class groups. The Nielsen-Thurston type of mapping classes is a commensurability invariant. One of the important facts of fibered commensurability on mapping class groups is for the case of pseudo-Anosovs, there is a unique minimal element in each fibered commensurability class. For outer automorphisms, we first show that being atoroidal and fully irreducible is a commensurability invariant. Then for such outer automorphisms, we prove that there is a unique minimal element in each fibered commensurability class, under a certain asymmetry condition on the ideal Whitehead graphs.