Normalizers and centralizers of cyclic subgroups generated by lone axis fully irreducible outer automorphisms

TL;DR

This paper characterizes the algebraic structure of centralizers and normalizers of cyclic subgroups generated by certain fully irreducible outer automorphisms, showing they are isomorphic to either infinite cyclic groups or specific free products.

Contribution

It establishes that the centralizer equals the stabilizer of the attracting lamination and is isomorphic to , and describes the normalizer as either  or a free product of two  groups.

Findings

Centralizer equals the stabilizer of the attracting lamination and is isomorphic to .

Normalizer is either  or  * .

Results apply to ageometric fully irreducible outer automorphisms with a unique axis.

Abstract

We let $\varphi$ be an ageometric fully irreducible outer automorphism so that its Handel-Mosher axis bundle consists of a single unique axis. We show that the centralizer $Cen(\langle\varphi\rangle)$ of the cyclic subgroup generated by $\varphi$ equals the stabilizer $\text{Stab}(\Lambda^+_\varphi)$ of the attracting lamination $\Lambda^+_{\varphi}$ and is isomorphic to $\mathbb Z$. We further show, via an analogous result about the commensurator, that the normalizer $N(\langle\varphi\rangle)$ of $\langle \varphi \rangle$ is isomorphic to either $\mathbb Z$ or $\mathbb Z_2 * \mathbb Z_2$.