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arXiv · 2107.08744

Generation and Simplicity in the Airplane Rearrangement Group

Matteo Tarocchi
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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].

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Matteo Tarocchi. 2023-10-23. Generation and Simplicity in the Airplane Rearrangement Group. https://doi.org/10.4171/ggd%2F772

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