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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.07577 |
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| _version_ | 1866908668805513216 |
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| author | Martin, Daniel E. |
| author_facet | Martin, Daniel E. |
| contents | 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 $24{,}504{,}480=2\text{lcm}(1,2,\dots,17)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Markoff triples and Nielsen equivalence in $\text{SL}_2(\mathbb{F}_p)$ Martin, Daniel E. Number Theory 11D25, 20H30, 37C85 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 $24{,}504{,}480=2\text{lcm}(1,2,\dots,17)$. |
| title | Markoff triples and Nielsen equivalence in $\text{SL}_2(\mathbb{F}_p)$ |
| topic | Number Theory 11D25, 20H30, 37C85 |
| url | https://arxiv.org/abs/2510.07577 |