Generation and Simplicity in the Airplane Rearrangement Group

TL;DR

This paper investigates the Airplane rearrangement group, proving it is finitely generated by Thompson groups, analyzing its commutator subgroup, and exploring its structural properties and subgroups.

Contribution

It establishes that the Airplane rearrangement group is finitely generated by Thompson groups and characterizes its commutator subgroup as simple and 2-transitive.

Findings

The group is generated by a copy of Thompson's F and T.

The abelianization of the group is isomorphic to Z.

The commutator subgroup is simple, finitely generated, and acts 2-transitively.

Abstract

We study the group $T_A$ of rearrangements of the Airplane limit space introduced by Belk and Forrest in [3]. We prove that $T_A$ is generated by a copy of Thompson's group $F$ and a copy of Thompson's group $T$, hence it is finitely generated. Then we study the commutator subgroup $[T_A, T_A]$, proving that the abelianization of $T_A$ is isomorphic to $\mathbb{Z}$ and that $[T_A, T_A]$ is simple, finitely generated and acts 2-transitively on the so-called components of the Airplane limit space. Moreover, we show that $T_A$ is contained in $T$ and contains a natural copy of the Basilica rearrangement group $T_B$ studied in [2].