greg mc

May 2021

- Markoff numbers and Frobenius conjecture
- Aigners monotonicity conjectures
- Group actions and labelling numbers
- Visualizing Markoff numbers

- Proofs from THE BOOK
- Convexity and Aigner's Conjectures
- Can I prove these with one figure ?

Markov numbers are integers that appear in triples which are solutions of a Diophantine equation the so-called Markov cubic

\(x^2 + y^2 + z^2 - 3x y z = 0.\)

\((1,1,1),(1,1,2),(1,2,5),(1,5,13)\)

\(\begin{pmatrix} 3 & -1 \\ 1 & 0 \end{pmatrix}\) is an automorph of \(x^2 + y^2 - 3x y.\)

so \(( v_n,v_{n+1},1)\) is a solution where

\(\begin{pmatrix}v_{n+1} \\ v_n \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ 1 & 0 \end{pmatrix}^n \begin{pmatrix}1 \\ 1 \end{pmatrix}\)

\(1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...\)

\((1,1,1),(1,1,2),(1,2,5),(1,5,13)\)

The largest integer in a triple determines the two other numbers.

m = Markoff number

- Jack Button for m prime
- Zhang An elementary proof...
- Baragar m, 3m - 2, 3m + 2 prime
- Bugeaud, Reutenauer, Siksek
- Multiplicities of simple closed geodesics
- Conclusion too hard!!!

- Markovâ€™s theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. 2013.

- M. Rabideaua, R. Schiffler, Continued fractions and orderings on the Markov numbers, Advances in Mathematics Vol 370, 2020.
- C Lagisquet and E. PelantovĂˇ and S. Tavenas and L. Vuillon, On the Markov numbers: fixed numerator, denominator, and sum conjectures.

There is a natural map (we'll see why shortly)

\(\mathbb{Q}\cup \infty \rightarrow\) Markoff numbers

\(p/q \mapsto m_{p,q}\)

Labeling Markoff numbers

- Markoff number = \(m_{p/q}\)
- Farey tree of coprime integers \(p,q\)
- Markoff tree of solutions to cubic
- Bass-Serre of a free product (\(PSL(2,\mathbb{Z})\))

- \(GL(2,\mathbb{Z})\) and its actions
- coprime integers \(p,q\)
- simple closed geodesics on a torus
- snake graphs
- "lengths"

coprime integers \(p,q\) live in a grid with a line

- \(\mathbb{Z}\) fundamental group of circle
- \(\mathbb{Z}^2\) fundamental group of torus
- \(\mathbb{Z}^2\) acting by translation on \(\mathbb{R}^2\).

- \(\mathbb{Z}^2\) acting by translation on \(\mathbb{R}^2\).
- infinitely many primitive elements
- \((a,b)\) primitive iff \(a,b \in \mathbb{Z}\) coprime

Next most interesting groups ?

- free product \(\mathbb{Z}*\mathbb{Z}\)
- automorphism group \(GL(2,\mathbb{Z})\)
- both lead to hyperbolic geometry

- \(GL(2,\mathbb{Z})\) acting by "base change" on \(\mathbb{Z}^2\)
- Bezout's identity \(\Rightarrow\) transitive on primitives
- Visualize \(GL(2,\mathbb{Z})\) action

\(\mathbb{Q}\cup \infty \subset\) circle/projective line

- \((a,b)\text{ primitive } \mapsto a/b \in \mathbb{Q}\cup \infty\)
- \(\begin{pmatrix} a & c \\ b & d \end{pmatrix} \mapsto\) arc joining \((a/b, c/d)\)
- \((a/b, c/d)\) are Farey neighbors

\(\mathbb{Q}\cup \infty \rightarrow\) Markoff numbers

\(p/q \mapsto m_{p,q}\)

- projective \(GL(2, \mathbb{Z})\) action on \(\mathbb{Q}\cup \infty\)
- action on Markoff numbers ?
- Vieta flipping

\(x^2 + y^2 + z^2 - 3x y z = 0.\)

- quadratic in \(x\), two roots \(x^\pm\)
- \(x^+ + x^- = 3yz\)
- involution \((x,y,z) \mapsto (3yz -x, y,z)\)

Peter Sarnak (Princeton and IAS)

Title: Strong approximation for Markoff surfaces

We discuss the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Markoff type affine cubic surfaces. These are dictated in part by the finite orbits of these actions on the algebraic points

Joint work with J.Bourgain and A.Gamburd.

- Vieta flips
- (cyclic) permutations of \(x,y,z\)
- get \(\mathbb{Z}/2 * \mathbb{Z}/3\) action
- = \(PSL(2,\mathbb{Z})\) action

Natural = \(PSL(2,\mathbb{Z})\)-equivariant map

\(\mathbb{Q}\cup \infty \rightarrow\) Markoff numbers \(p/q \mapsto m_{p/q}\)

- \((1:1) \mapsto 1/1 \mapsto 2\)
- \((0:1) \mapsto 0/1 \mapsto 1\)
- \((1:0) \mapsto \infty \mapsto 1\)
- actions = projective on left and by autos on right

comes from Bass-Serre tree of \(PSL(2,\mathbb{Z})\)

- so we have seen the structure of the Markoff numbers
- is there a bette way?
- is there hidden information?

\(x^2 + y^2 + z^2 - x y z = 0.\)

- positive characteristic Sarnak et al.
- over \(\mathbb{C}\) ?
- Bowditch
- Sakuma et al.
- SP Tan et al.

- \(a,b\) generate \(\mathbb{Z}*\mathbb{Z}\)
- coprime \((p,q) \mapsto w \in \langle a, b \rangle\)
- \(1/c (|p| + |q|) \leq \ell_w \leq c(|p| + |q|)\)

\(\rho, \gamma \mapsto \ell_\gamma(\rho)\) is a continuous map

\(\mathcal{T}(\Sigma) \times \mathcal{ML} \rightarrow \mathbb{R}\)

- Generalisation Brock
- Question how regular is it?