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Type systems and maximal subgroups of Thompson's group $V$

James Belk, Collin Bleak, Martyn Quick, Rachel Skipper

TL;DR

It is shown that there are uncountably many maximal subgroups of V that occur as the stabilizers of simple type systems and do not arise in the form of previously known maximal sub groups.

Abstract

We introduce the concept of a type system~$\Part$, that is, a partition on the set of finite words over the alphabet~$\{0,1\}$ compatible with the partial action of Thompson's group~$V$, and associate a subgroup~$\Stab{V}{\Part}$ of~$V$. We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of~$V$. We also find an uncountable family of pairwise non-isomorphic maximal subgroups of~$V$. These maximal subgroups occur as stabilizers of infinite simple type systems and have not been described in previous literature: specifically, they do not arise as stabilizers in $V$ of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of $V$ (both related to primitivity) are each satisfied only by $V$ itself, giving new ways to recognise when a subgroup of $V$ is not actually proper.

Type systems and maximal subgroups of Thompson's group $V$

TL;DR

It is shown that there are uncountably many maximal subgroups of V that occur as the stabilizers of simple type systems and do not arise in the form of previously known maximal sub groups.

Abstract

We introduce the concept of a type system~, that is, a partition on the set of finite words over the alphabet~ compatible with the partial action of Thompson's group~, and associate a subgroup~ of~. We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of~. We also find an uncountable family of pairwise non-isomorphic maximal subgroups of~. These maximal subgroups occur as stabilizers of infinite simple type systems and have not been described in previous literature: specifically, they do not arise as stabilizers in of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of (both related to primitivity) are each satisfied only by itself, giving new ways to recognise when a subgroup of is not actually proper.
Paper Structure (16 sections, 57 theorems, 18 equations, 9 figures)

This paper contains 16 sections, 57 theorems, 18 equations, 9 figures.

Table of Contents

  1. Acknowledgements
  2. Background
  3. Stabilizers of finite sets of fixed tail class
  4. Labelling subgroups of V
  5. The semigroup associated to the equivalence relation
  6. Simple type systems
  7. Maximality of stabilizer subgroups
  8. Description of the stabilizers of simple type systems
  9. Atomic multinuclear type systems
  10. Atomic quasinuclear type systems
  11. Nuclear type systems
  12. Uncountably many distinct maximal subgroups.
  13. Uncountably many isomorphism types
  14. Conditions which prevent maximality
  15. Remark
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\mathcal{P}$ be a finite simple type system on $\Omega$. Then $\operatorname{Stab}_{V}(\mathcal{P})$ is a maximal subgroup of $V$.

Figures (9)

  • Figure 1: Label diagram for Example \ref{['ex:stabilizerZero']}
  • Figure 2: Label diagram for Example \ref{['ex:SlopeFour']}
  • Figure 3: Label diagram for a particular multinuclear type system
  • Figure 4: Label diagram for the partition $\{0\mathfrak{C},10\mathfrak{C},11\mathfrak{C}\}$, which has stabilizer isomorphic to $V^3\rtimes S_3$.
  • Figure 5: Label diagram for a type system whose stabilizer is the stabilizer in $V$ of the two-point set $\bigl\{\overline{01},\overline{10}\bigr\}$.
  • ...and 4 more figures

Theorems & Definitions (94)

  • Theorem 1
  • Theorem 1
  • Theorem 1
  • Theorem 1
  • Theorem 1
  • Theorem 1
  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • Proposition 2.1: DixonMort
  • ...and 84 more