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Markoff triples and Nielsen equivalence in $\text{SL}_2(\mathbb{F}_p)$

Daniel E. Martin

TL;DR

The paper develops a deep connection between Nielsen moves in $\text{SL}_2(\mathbb{F}_p)$ and Markoff-type equations, formalizing the invariant space $\mathscr{P}(\mathbb{F}_p)$ and a polynomial reduction operator $\Phi$ to study Markoff-curve orbits. It proves the $Q$-Classification Conjecture for primes $p$ with $24{,}504{,}480 mid p^2-1$, by showing a single Γ-orbit on Markoff triples for all $\kappa \in \mathbb{F}_p\setminus\{4\}$ except a short list of exceptional cases, and deduces the Classification and $T$-Classification conjectures in this setting. The general reduction to a matrix rank problem is made precise: for any $d$, the conjecture reduces to invertibility of a roughly $2d\times 2d$ matrix over $\mathbb{Q}[\kappa]$, and the authors verify invertibility for all prime powers $d\le 17$, yielding the modulus $2\operatorname{lcm}(1,\dots,17)$. A comprehensive framework combining Markoff theory, eigenvector constructions, and computational algebra (Sage) is developed to bound dimensions of $\mathscr{P}_{2n}(\mathbb{F}_p,d)$ and to confirm the conjectures in broad arithmetic regimes. The results bridge Markoff-type dynamics with Nielsen equivalence classes in linear groups, providing both theoretical and computational evidence for the conjectures across a wide congruence spectrum.

Abstract

In 2013, Darryl McCullough and Marcus Wanderley made a series of conjectures that describe the Nielsen equivalence classes and $T_2$-equivalence classes of pairs of generators for $\text{SL}_2(\mathbb{F}_q)$ and the Markoff equivalence classes of triples in $\mathbb{F}_q^3$ that solve $x^2+y^2+z^2=xyz+κ$ for some $κ\in\mathbb{F}_q$. (The case $κ=0$ was originally conjectured by Baragar in 1991.) We prove that one of the McCullough-Wanderley conjectures, the "Q-Classification Conjecture" on Markoff triples, implies the others. Then we prove that the Q-Classification Conjecture holds if $q=p$ is a prime such that $24{,}504{,}480$ does not divide $p^2-1$. More generally, for any integer $d$, we reduce the Q-Classification Conjecture for all primes $p\not\equiv \pm 1\,\text{mod}\,d$ to checking whether a roughly $2d\times 2d$ matrix with entries in $\mathbb{Q}[κ]$ is invertible. We (and SageMath) perform this invertibility check for all prime powers $d$ up to $17$, hence the modulus $24{,}504{,}480=2\text{lcm}(1,2,\dots,17)$.

Markoff triples and Nielsen equivalence in $\text{SL}_2(\mathbb{F}_p)$

TL;DR

The paper develops a deep connection between Nielsen moves in and Markoff-type equations, formalizing the invariant space and a polynomial reduction operator to study Markoff-curve orbits. It proves the -Classification Conjecture for primes with , by showing a single Γ-orbit on Markoff triples for all except a short list of exceptional cases, and deduces the Classification and -Classification conjectures in this setting. The general reduction to a matrix rank problem is made precise: for any , the conjecture reduces to invertibility of a roughly matrix over , and the authors verify invertibility for all prime powers , yielding the modulus . A comprehensive framework combining Markoff theory, eigenvector constructions, and computational algebra (Sage) is developed to bound dimensions of and to confirm the conjectures in broad arithmetic regimes. The results bridge Markoff-type dynamics with Nielsen equivalence classes in linear groups, providing both theoretical and computational evidence for the conjectures across a wide congruence spectrum.

Abstract

In 2013, Darryl McCullough and Marcus Wanderley made a series of conjectures that describe the Nielsen equivalence classes and -equivalence classes of pairs of generators for and the Markoff equivalence classes of triples in that solve for some . (The case was originally conjectured by Baragar in 1991.) We prove that one of the McCullough-Wanderley conjectures, the "Q-Classification Conjecture" on Markoff triples, implies the others. Then we prove that the Q-Classification Conjecture holds if is a prime such that does not divide . More generally, for any integer , we reduce the Q-Classification Conjecture for all primes to checking whether a roughly matrix with entries in is invertible. We (and SageMath) perform this invertibility check for all prime powers up to , hence the modulus .

Paper Structure

This paper contains 17 sections, 50 theorems, 212 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. Background
  4. Relations among the McCullough--Wanderley conjectures
  5. Overview of the remaining sections
  6. The main definitions
  7. Proof strategy
  8. The conjectured space $\mathscr{P}(\mathbb{F}_p)$
  9. Properties of $\Phi$ and $\Phi_x$
  10. Degree bounds and the canonical form
  11. Partial formulas for $\Phi(f)$
  12. Eigenvectors for $\alpha\neq\pm 2$
  13. Eigenvector existence
  14. Another partial formula
  15. The space $\mathscr{P}_{\!x}(R,d)$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $p$ be prime such that $24{,}504{,}480\centernot\mid p^2-1$. For any $\kappa\in\mathbb{F}_p\backslash\{4\}$, there is a single orbit of solutions in $\mathbb{F}_p^3$ to $x^2+y^2+z^2=xyz+\kappa$ under the group generated by other than the following exceptional orbits up to coordinate permutation, provided the elements exist in $\mathbb{F}_p$:

Theorems & Definitions (102)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3: "Classification Conjecture" wanderley
  • Conjecture 1.4: "$Q$-Classification Conjecture" wanderley
  • Theorem 1.5
  • Proposition 2.1: Proposition 5.2 and Lemma 5.3 in wanderley
  • Lemma 2.2: Lemma 2.1(ii) in wanderley
  • Theorem 2.3: Theorem 3 in macbeath
  • Lemma 2.4
  • proof
  • ...and 92 more