Markoff numbers

greg mc

May 2021

Geometry of discrete groups and hyperbolic spaces

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

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

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

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

infinity of Markoff numbers

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

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

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

Odd index Fibonacci numbers are Markoff numbers

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

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

Frobenius uniqueness conjecture

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

Partial results

m = Markoff number

Aigner's monotonicity conjectures

  • Markov’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. Pelantová 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

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

A tale of three trees

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 (\(PSL(2,\mathbb{Z})\))

Story of correspondences

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

Grids

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

Matchings

Groups

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

Group actions

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

Two choices (red/blue pill)

Next most interesting groups ?

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

Group actions

  • \(GL(2,\mathbb{Z})\) 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

source

source

natural map ?

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

Vieta flips/involutions

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

  • 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.

Automorphisms

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

Tree structure

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

Vizualization

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

Motivation

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

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

Fibonacci growth

Bowditch

Fibonacci growth

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

Continuity of length

\(\rho, \gamma \mapsto \ell_\gamma(\rho)\) is a continuous map
\(\mathcal{T}(\Sigma) \times \mathcal{ML} \rightarrow \mathbb{R}\)

Is there a norm hidden here?