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Generation and Simplicity in the Airplane Rearrangement Group

Matteo Tarocchi

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].

Generation and Simplicity in the Airplane Rearrangement Group

Abstract

We study the group of rearrangements of the Airplane limit space introduced by Belk and Forrest in [3]. We prove that is generated by a copy of Thompson's group and a copy of Thompson's group , hence it is finitely generated. Then we study the commutator subgroup , proving that the abelianization of is isomorphic to and that is simple, finitely generated and acts 2-transitively on the so-called components of the Airplane limit space. Moreover, we show that is contained in and contains a natural copy of the Basilica rearrangement group studied in [2].

Paper Structure

This paper contains 23 sections, 20 theorems, 15 equations, 30 figures.

Table of Contents

  1. Thompson's groups F and T
  2. Limit spaces and rearrangements
  3. Replacement systems and limit spaces
  4. Cells and rearrangements
  5. The rearrangement group of the Basilica limit space
  6. The Airplane limit space and its rearrangements
  7. Components of the Airplane limit space
  8. Rays of the Airplane limit space
  9. The elements of TA
  10. Component paths
  11. Copies of Thompson's groups F and T in TA
  12. A copy of Thompson's group T in TA
  13. A copy of Thompson's group F in TA
  14. Generators of TA
  15. The commutator subgroup of TA
  16. ...and 8 more sections

Key Result

Theorem \oldthetheorem

$rist(C_0) = \langle \beta, \gamma, \delta \rangle \simeq T$, and it acts on $\partial C_0$ as $T$ does on $S^1$. In particular, its action is 2-transitive on the set of central rays.

Figures (30)

  • Figure 1: The Airplane limit space.
  • Figure 2: An example of dyadic subdivision of $[0,1]$.
  • Figure 3: The generators $X_0$ and $X_1$ of Thompson's group $F$.
  • Figure 4: The three generators of Thompson's group $T$.
  • Figure 5: The Airplane replacement system $\mathcal{A}$.
  • ...and 25 more figures

Theorems & Definitions (34)

  • Definition \oldthetheorem
  • Definition \oldthetheorem: Definition 1.14(2) of belk2016rearrangement)
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • ...and 24 more