Generation and simplicity in the airplane rearrangement group

Abstract

We study the group $T_A$ of rearrangements of the airplane limit space introduced by Belk and Forrest (2019). 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 by Belk and Forrest (2015).