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Tree Pairs for Algebraic Bieri-Strebel Groups

Lewis Molyneux

TL;DR

This work investigates whether the tree-pair representations central to Thompson-like groups extend to algebraic Bieri-Strebel groups beyond the quadratic case. Building on subdivision-polynomial methods and caret-type analysis, it proves that the algebraic Bieri-Strebel group $G([0.1], \mathbb{Z}[β], \langle β\rangle)$ with $0<β<1$ as the root of $a_n x^{2n}+\cdots+ a_1 x^n-1$ cannot admit a well-defined tree-pair representation, illustrating obstructions that persist at higher degrees. The results extend Winstone's quadratic theory by providing explicit higher-degree counterexamples (e.g., using $P(x)=x^4+x^2-1$) and a lemma that $\,\sqrt{β}\notin \mathbb{Z}[β]$ for quadratic subdivision polynomials, which propagates nonrepresentability to groups like $F_{\sqrt[2n]{β}}$. Together with a conjecture concerning non-coprime power polynomials, the paper clarifies limits of the tree-pair framework for large classes of Bieri-Strebel groups and informs our understanding of related finiteness properties in Thompson-like groups.

Abstract

We reintroduce a previously discovered method for constructing tree pair representations for Algebraic Bieri-Strebel groups, as well as demonstrate a class of higher order groups that cannot have a tree pair representation. In doing so, we demonstrate that there is no maximum degree such that for all polynomials of higher degree, the associated Algebraic Bieri Strebel group must have a tree-pair representation.

Tree Pairs for Algebraic Bieri-Strebel Groups

TL;DR

This work investigates whether the tree-pair representations central to Thompson-like groups extend to algebraic Bieri-Strebel groups beyond the quadratic case. Building on subdivision-polynomial methods and caret-type analysis, it proves that the algebraic Bieri-Strebel group with as the root of cannot admit a well-defined tree-pair representation, illustrating obstructions that persist at higher degrees. The results extend Winstone's quadratic theory by providing explicit higher-degree counterexamples (e.g., using ) and a lemma that for quadratic subdivision polynomials, which propagates nonrepresentability to groups like . Together with a conjecture concerning non-coprime power polynomials, the paper clarifies limits of the tree-pair framework for large classes of Bieri-Strebel groups and informs our understanding of related finiteness properties in Thompson-like groups.

Abstract

We reintroduce a previously discovered method for constructing tree pair representations for Algebraic Bieri-Strebel groups, as well as demonstrate a class of higher order groups that cannot have a tree pair representation. In doing so, we demonstrate that there is no maximum degree such that for all polynomials of higher degree, the associated Algebraic Bieri Strebel group must have a tree-pair representation.
Paper Structure (8 sections, 6 theorems, 20 equations, 10 figures)

This paper contains 8 sections, 6 theorems, 20 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Thompson's group
  3. Tree Pairs for F
  4. $\textbf{F}_{n}$.
  5. Bieri-Strebel Groups
  6. Algebraic Numbers and Subdivision Polynomials
  7. Winstone's Tree Pairs
  8. Higher Order Polynomials

Key Result

Theorem 1.1

The algebraic Bieri-Strebel group $G([0.1], \mathbb{Z}[\beta], \langle \beta \rangle)$, where $\beta$ is the root of the polynomial $ax^{2n}+bx^n-1$ with $0 < \beta <1$, cannot have a tree-pair representation.

Figures (10)

  • Figure 1: The same element of $\textbf{F}$, expressed as both a function $f:I \rightarrow I$ and as a pair of partitions.
  • Figure 2: The element from \ref{['x01']}, now presented as a tree-pair. Note how the nodes in the trees are positioned similarly to the breakpoints in the partitions.
  • Figure 3: A tree-pair diagram with a redundant caret, highlighted in red, and the equivalent reduced diagram.
  • Figure 4: Demonstration of composition of tree-pairs.
  • Figure 5: A partition of the unit interval based on the subdivision polynomial $x^2 + 2x -1$
  • ...and 5 more figures

Theorems & Definitions (19)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Conjecture 3.8
  • Theorem 4.1
  • ...and 9 more