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Strong Approximation for the Character Variety of the Four-Times Punctured Sphere

Nathaniel Kingsbury-Neuschotz

Abstract

We study the orbits of the solutions to the Markoff-type equation $$X^2 + Y^2 + Z^2 = AX + BY + CZ + D$$ in $\mathbb{F}_p$ for fixed integers $A, B, C,$ and $D$ under the group of symmetries $Γ$ generated by $$V_1: (x, y, z)\mapsto (A + yz - x, y, z),$$ $$V_2: (x, y, z)\mapsto (x, B + xz - y, z),\text{ and}$$ $$V_3: (x, y, z)\mapsto (x, y, C + xy - z).$$ For most quadruples of parameters $(A, B, C, D)$, we show that there is a density one set of primes $p$ such that $Γ$ acts transitively on the bulk of the solutions mod $p$, with the remainder breaking up into a few small orbits which arise from finite orbits within the solutions over $\mathbb{C}$. For those ``degenerate'' quadruples of parameters $(A, B, C, D)$ to which this result does not apply, we show that there must be at least 2 large orbits, and in some cases 4 large orbits, under the action of this group. Our results become especially interesting when applied to two special subfamilies. The first is $$X^2 + Y^2 + Z^2 = XYZ + k$$ for $k \neq 4$, which arises in the study of the combinatorial group theory of $\text{SL}_2(\mathbb{F}_p)$. Our results very nearly prove the $Q$-classification conjecture of McCullough and Wanderley for density 1 of all primes, and thus by the work of Martin very nearly proves their Classification and $T$-Classification conjectures for density 1 of all primes. The second special family is $$x_1^2 + x_2^2 + x_3^2 + a_1x_2x_3 + a_2x_1x_3 + a_3x_1x_2 = (3+a_1+a_2+a_3)x_1x_2x_3,$$ which arises from certain generalized cluster algebras. Here, our notion of ``degenerate'' parameters $(A, B, C, D)$ specializes to the degeneracy condition of de Courcy-Ireland, Litman, and Mizuno. For this family, their results imply that our transitivity result applies to all sufficiently large primes $p$, independent of $a_1, a_2,$ and $a_3.$

Strong Approximation for the Character Variety of the Four-Times Punctured Sphere

Abstract

We study the orbits of the solutions to the Markoff-type equation in for fixed integers and under the group of symmetries generated by For most quadruples of parameters , we show that there is a density one set of primes such that acts transitively on the bulk of the solutions mod , with the remainder breaking up into a few small orbits which arise from finite orbits within the solutions over . For those ``degenerate'' quadruples of parameters to which this result does not apply, we show that there must be at least 2 large orbits, and in some cases 4 large orbits, under the action of this group. Our results become especially interesting when applied to two special subfamilies. The first is for , which arises in the study of the combinatorial group theory of . Our results very nearly prove the -classification conjecture of McCullough and Wanderley for density 1 of all primes, and thus by the work of Martin very nearly proves their Classification and -Classification conjectures for density 1 of all primes. The second special family is which arises from certain generalized cluster algebras. Here, our notion of ``degenerate'' parameters specializes to the degeneracy condition of de Courcy-Ireland, Litman, and Mizuno. For this family, their results imply that our transitivity result applies to all sufficiently large primes , independent of and
Paper Structure (17 sections, 29 theorems, 216 equations, 4 figures)

This paper contains 17 sections, 29 theorems, 216 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Equivalence of Parameters
  3. Extra Automorphisms
  4. Outline of the Paper
  5. Finite Orbits in Characteristic Zero
  6. Applications to Group Theory
  7. Generalized Cluster Algebras and the Nondegeneracy Condition
  8. Analysis of the Conic Sections
  9. Comparison of Gamma and Gamma'
  10. Endgame
  11. Connectivity of the Cage
  12. Middlegame
  13. Opening
  14. Obstruction to Transitivity for Degenerate Parameters
  15. Acknowledgements
  16. ...and 2 more sections

Key Result

Theorem 1.1

Suppose that $(A, B, C, D)$ is a quadruple of integers which is nondegenerate in the sense of definition nondegeneracyCondition. Then there is a density one set of primes $p$ for which $S_{A, B, C, D}(\mathbb{F}_p)$ consists of a single giant orbit under $\Gamma,$ together with small orbits that are

Figures (4)

  • Figure 1: The Markoff Tree
  • Figure 2: Exceptional finite orbit for the parameters $(A, B, C, D) = (0, -1, -1, 0)$ starting from $(1, -1, -1)$
  • Figure 3: Exceptional finite orbit for the parameters $(A, B, C, D) = (0, 0, 0, 3)$ starting from $(1, \sqrt{2}, \sqrt{2})$
  • Figure 4: Exceptional finite orbit for the parameters $(A, B, C, D) = (\sqrt{2}, 0, 0, 1)$ starting from $(\sqrt{2}, -1, -\sqrt{2})$

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1.2: Uniformity in $A, B, C,$ and $D$
  • Definition 1.3
  • Definition 1.4
  • Proposition 2.1: LT, Lemma 39 part 1
  • Theorem 2.2: LT, Theorem 1
  • Remark 2.3
  • Theorem 2.5
  • Theorem 3.1: MartinBigPaper, Theorem 1.5
  • Theorem 4.1: pDivisibilityClusterAlg, Theorem 1.1
  • ...and 43 more