Geodesic
Generation and simplicity in the airplane rearrangement group
Matteo Tarocchi1 orcid
1Universitá degli Studi di Firenze, Firenze, Italy; Università degli Studi di Milano-Bicocca, Milano, Italy
Groups, geometry, and dynamics, Tome 18 (2024) no. 2, pp. 603-634

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DOI

We study the group TA​ of rearrangements of the airplane limit space introduced by Belk and Forrest (2019). We prove that TA​ 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 [TA​,TA​], proving that the abelianization of TA​ is isomorphic to Z and that [TA​,TA​] 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 TB​ studied by Belk and Forrest (2015).
DOI : 10.4171/ggd/772
Classification : 20F65, 20E32, 20F05, 20F38, 28A80
Mots-clés : Thompson’s groups, Thompson-like, fractal, Julia set, airplane Julia set, airplane fractal

Matteo Tarocchi  1

1 Universitá degli Studi di Firenze, Firenze, Italy; Università degli Studi di Milano-Bicocca, Milano, Italy 
Matteo Tarocchi. Generation and simplicity in the airplane rearrangement group. Groups, geometry, and dynamics, Tome 18 (2024) no. 2, pp. 603-634. doi: 10.4171/ggd/772
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