Generation and simplicity in the airplane rearrangement group.

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 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 TA is contained in T and contains a natural copy of the basilica rearrangement group T_B studied by Belk and Forrest (2015). [ABSTRACT FROM AUTHOR]