Generation and simplicity in the airplane rearrangement group

We study the group T₀ of rearrangements of the airplane limit space introduced by Belk and Forrest (2019). We prove that T₀ 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 T₀, T₀, proving that the abelianization of T₀ is isomorphic to Z and that T₀, T₀ is simple, finitely generated and acts 2-transitively on the so-called components of the airplane limit space. Moreover, we show that T₀ is contained in T and contains a natural copy of the basilica rearrangement group T₁ studied by Belk and Forrest (2015).