Generation and Simplicity in the Airplane Rearrangement Group
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929328108863488 |
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| author | Tarocchi, Matteo |
| author_facet | Tarocchi, Matteo |
| contents | 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2107_08744 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Generation and Simplicity in the Airplane Rearrangement Group Tarocchi, Matteo Group Theory 20F65 (Primary) 20F38, 28A80, 20E32, 20F05 (Secondary) 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]. |
| title | Generation and Simplicity in the Airplane Rearrangement Group |
| topic | Group Theory 20F65 (Primary) 20F38, 28A80, 20E32, 20F05 (Secondary) |
| url | https://arxiv.org/abs/2107.08744 |