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

Hidetoshi Masai, Ryosuke Mineyama

Published 2017-05-09Version 1

We define and discuss fibered commensurability of outer automorphisms of the free groups, which lets us study symmetries 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. In this paper, we first show that being ageometric 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 asymmetry assumption on the ideal Whitehead graphs.