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The Maximality of $T$ in Thompson's group $V$

James Belk, Collin Bleak, Martyn Quick, Rachel Skipper

TL;DR

The paper proves that Thompson's group $T$ is a maximal subgroup of Thompson's group $V$ in the natural action on Cantor space $\mathfrak{C}$. It develops a framework based on prefix-substitution swaps, interleaved permutations, and tree-pair representations to control elements of $V$ and relate them to $T$. The main result is a complete maximality proof, built around the Interleaved Shaping Lemma and related conjugation techniques to show that any outside element together with $T$ generates $V$. This work clarifies the subgroup lattice of Thompson groups and demonstrates how dynamical properties on the Cantor space translate into concrete algebraic generation results.

Abstract

We show that R. Thompson's group $T$ is a maximal subgroup of the group $V$. The argument provides examples of foundational calculations which arise when expressing elements of $V$ as products of transpositions of basic clopen sets in Cantor space $\mathfrak{C}$.

The Maximality of $T$ in Thompson's group $V$

TL;DR

The paper proves that Thompson's group is a maximal subgroup of Thompson's group in the natural action on Cantor space . It develops a framework based on prefix-substitution swaps, interleaved permutations, and tree-pair representations to control elements of and relate them to . The main result is a complete maximality proof, built around the Interleaved Shaping Lemma and related conjugation techniques to show that any outside element together with generates . This work clarifies the subgroup lattice of Thompson groups and demonstrates how dynamical properties on the Cantor space translate into concrete algebraic generation results.

Abstract

We show that R. Thompson's group is a maximal subgroup of the group . The argument provides examples of foundational calculations which arise when expressing elements of as products of transpositions of basic clopen sets in Cantor space .
Paper Structure (3 sections, 5 theorems, 13 equations)

This paper contains 3 sections, 5 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The argument

Key Result

Lemma 3.2

Let $(\alpha_i)_{i=1}^k$ and $(\beta_i)_{i=1}^k$ be ordered, interleaved sequences of $k$ leaves on some finite rooted binary trees $A$ and $B$ respectively. Then there are finite rooted binary trees $D$ and $R$ with the same number of leaves and a bijection $\tau$ from the leaves of $D$ to the leav

Theorems & Definitions (10)

  • Example 3.1
  • Lemma 3.2: The Interleaved Shaping Lemma
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Lemma 3.6
  • proof
  • Theorem 3.7
  • proof