[Paper Review] On Jones Subgroup of R. Thompson's Group $T$

Jordan Nikkel, Yunxiang Ren|arXiv (Cornell University)|Oct 19, 2017

Advanced Operator Algebra Research4 references3 citations

TL;DR

This paper fully characterizes Jones' subgroup $τ T$ of Thompson's group $T$, proving it coincides with its commensurator in $T$, which implies the associated unitary representation is irreducible. It provides a finite presentation for $τ T$ as an annular diagram group and shows it is not isomorphic to $T_3$, despite similar presentations via Tietze transformations.

ABSTRACT

Jones introduced unitary representations for the Thompson groups $F$ and $T$ from a given subfactor planar algebra. Some interesting subgroups arise as the stabilizer of certain vector, in particular the Jones subgroups $\vec{F}$ and $\vec{T}$. Golan and Sapir studied $\vec{F}$ and identified it as a copy of the Thompson group $F_3$. In this paper we completely describe $\vec{T}$ and show that $\vec{T}$ coincides with its commensurator in $T$, implying that the corresponding unitary representation is irreducible. We also generalize the notion of the Stallings 2-core for diagram groups to $T$, showing that $\vec{T}$ and $T_3$ are not isomorphic, but as annular diagram groups they have very similar presentations.

Motivation & Objective

• To fully characterize Jones' subgroup $τ T$ of Thompson's group $T$ using diagram group theory and planar algebra representations.
• To prove that $τ T$ coincides with its commensurator in $T$, implying irreducibility of the associated unitary representation.
• To generalize the Stallings 2-core construction from $F$ to $T$ for subgroup analysis.
• To provide a finite presentation of $τ T$ as an annular diagram group.
• To show that $τ T$ and $T_3$ are not isomorphic, despite having presentations related by Tietze transformations.

Proposed method

• Use Thompson graphs to define and analyze $τ T$, leveraging the bipartite condition for membership.
• Construct the Stallings 2-core for subgroups of $T$, generalizing the method from $F$ to annular diagram groups.
• Apply the 2-core construction to $τ T$ to show it is equal to its own core, implying a finite presentation as an annular diagram group.
• Use the core structure to derive a finite presentation for $τ T$ in terms of generators and relations.
• Compare $τ T$ and $T_3$ via their annular diagram group presentations and analyze group-theoretic invariants like element orders.
• Use parity arguments on dyadic rationals and tree pair representations to prove non-isomorphism between $τ T$ and $T_3$.

Experimental results

Research questions

• RQ1 Does Jones' subgroup $τ T$ coincide with its commensurator in $T$, and what does this imply for the associated unitary representation?
• RQ2 Can the Stallings 2-core construction be generalized from $F$ to $T$, and how does it help analyze subgroups of $T$?
• RQ3 What is a finite presentation of $τ T$ as an annular diagram group?
• RQ4 Are $τ T$ and $T_3$ isomorphic, despite having similar presentations via Tietze transformations?
• RQ5 What is the precise relationship between $τ T$ and the dyadic parity of binary digits of dyadic rationals in $[0,1)$?

Key findings

• Jones' subgroup $τ T$ coincides with its commensurator in $T$, which implies that the associated unitary representation of $T$ is irreducible.
• The subgroup $τ T$ is precisely the set of all elements in $T$ that either preserve or switch the parity of the sum of digits of all dyadic rationals in $[0,1)$ when written in binary.
• The Stallings 2-core of $τ T$ is equal to $τ T$ itself, which provides a finite presentation of $τ T$ as an annular diagram group.
• The group $τ T$ admits a finite presentation as an annular diagram group: $τ T = π^a(σ, au)$ with relations $e = ff$, $f = fe$, and base generator $e$, where $σ, au$ are specific diagram words.
• Although $τ T$ and $T_3$ have presentations that differ only by a Tietze transformation, they are not isomorphic, as $T_3$ contains no element of order two while $τ T$ does.
• The proof of non-isomorphism relies on a parity argument on the number of leaves in ternary tree pair representations, showing a contradiction if an element of order two existed in $T_3$.