Markoff numbers

greg mc shane

• slides : google greg mcshane github
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Markoff numbers are integers that appear a Markoff triple

which are solutions of a Diophantine equation
the so-called Markoff cubic

Odd index Fibonacci numbers are Markoff numbers

Odd index Pell numbers are Markoff numbers

Frobenius uniqueness conjecture

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

Partial results

m = Markoff number

• Jack Button for m prime
• Baragar m, 3m - 2, 3m + 2 prime
• Zhang An elementary proof...
• Lang, Tan A simple proof....
• Bugeaud, Reutenauer, Siksek
• Conclusion too hard!!!

Martin Aigner

• Proofs from THE BOOK
• Convexity and Aigner's Conjectures
• Prove his conjectures with one figure?

Aigner's monotonicity conjectures

• Markoff’s theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. 2013.
• M. Rabideau, R. Schiffler,
  Continued fractions and orderings on the Markoff numbers,
  Advances in Mathematics Vol 370, 2020. published
• C Lagisquet and E. Pelantová and S. Tavenas and L. Vuillon, On the Markoff numbers: fixed numerator, denominator, and sum conjectures. published

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

Aigner's conjectures proof

Labeling Markoff numbers

A tale of three trees

• Markoff number =
• Farey "tree" of coprime integers
• Markoff tree of solutions to the cubic
• Bass-Serre of a free product

coprime integers

• arc on a punctured torus
• closed geodesic (disjoint from the arc)
• snake graph and its perfect matchings
• "lengths" that verify a Ptolemy inequality

Group actions

circle/projective line

• arc joining
• are Farey neighbors iff
• obvious transitive action on Farey neighbors

source

source

natural map ?

Markoff numbers

• action on
• action on Markoff numbers/triples ?
• Vieta jumping

Vieta jumps/flips/involutions

• quadratic in , two roots
• Vieta formula
• involution

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

• Vieta flips
• (cyclic) permutations of
• action of

Natural = -equivariant map

• actions = projective on left and by autos on right

Tree structure

comes from Bass-Serre tree of

Uniqueness conjecture

• The largest integer in a triple determines the two other numbers.
• The multiplicity of any number in the complementary regions to the tree is at most 6

Modern theory: H. Cohn

Approach to Markoff’s Minimal Forms Through Modular Functions (1955)

• modular torus = quotient of upper half plane by commutator subgroup of , acting by Mobius transformations
• relates Markoff numbers to lengths of simple closed geodesics

• modular torus = quotient of upper half plane by commutator subgroup of
• obtained from a pair of ideal triangles by identification
• elliptic involution swaps triangles fixes midpoint of diagonal

Character variety

modular torus =

• fundamental group of the torus.
• any hyperbolic torus = ,
• discrete faithful representation
• lifts to
• generators of
• Definition character map

Puncture condition

is a loop round the puncture
so its holonomy is parabolic and in fact:

• ie Markoff cubic up to a change of variable

Cohn (and others) shows that the

• permutations
• the Vieta flips

used to construct Markoff's binary tree are induced by automorphisms of the fundamental group of the torus.

Exo

• Nielsen move Vieta flip

Theorem

The semi-algebraic set:

can be identified with the Teichmueller space of the punctured torus.

Counting problem

Theorem

• Zagier (1982) On the Number of Markoff Numbers Below a Given Bound.
• Greg McShane, Igor Rivin (1995) A norm on homology of surfaces and counting simple geodesics

Counting closed simple geodesics

• character map

• is generator iff essential simple closed curve representing its conjugacy/free homotopy class

Simple representatives

• blue curve is simple representative of its homotopy class
• not every homotopy class contains a simple curve
• every (non trivial) homology class has a representative that is a (multiple) of a simple curve

Norms and minimizers

Let be an essential closed curve its length.

• convexity/triangle inequality
• any pair of curves in linearly independent homology classes intersect
• a curve with self intersections is never a minimizer

• Cluster algebra folks call this a smoothing

Corollary: Let T be a punctured torus with a hyperbolic structure.

• Then, the shortest multicurve representing a non-trivial homology class is a simple closed geodesic if is a primitive homology class, and a multiply covered geodesic otherwise.
• In addition, the shortest multicurve representing is unique.

Unit ball

Unit ball and counting

• the area of the unit ball depends on the hyperbolic structure
• with Rivin we studied it, but now it's called the Mirzakhani function :(

Why log ?

• important monotone increasing on

Aigner's conjectures

Reformulate Aigner's conjectures

Markoff number
monotone increasing on

• Let be real non negative numbers and then

• If in addition then
  

Aigner's conjectures proof

• Gaster

source

On the ordering of the Markoff numbers
Kyungyong Lee, Li Li, Michelle Rabideau, Ralf Schiffler

The proof uses a connection to cluster algebras. It was observed
in [P, BBH] that the Markoff numbers can be obtained from the cluster variables in the cluster
algebra of the once-punctured torus by specializing the initial cluster variables to 1. Moreover, the clusters in the cluster algebra then specialize to the Markoff triples. On the other hand, the cluster variables can be computed by a combinatorial formula given as a summation over the perfect matchings of a so-called snake graph.

Snake graph

Button's Theorem

If is a Markoff number which is prime
then there is a unique triple

Theorem (Fermat)

Let be a prime then has a solution over

• iff or is a multiple of 4.

• Button's theorem follows from "unicity" of

• unique factorisation

Frobenius uniqueness conjecture

• The multiplicity of any number in the complementary regions to the tree is at most 6

• modular torus
• loop around the cusp
• automorphism group
• "generator" of the automorphism group is
• normalises so induces an involution of
• the involution has 3 fixed points which lift to the orbit of i.

• elliptic involution swaps triangles fixes midpoint of diagonal
• normalises induces an involution of
• the fixed points lift to the

Ford circles

Definitions

arc = Poincaré geodesic joining

• -length of arc

• -length of arc on is the length of a lift to

• acts by Mobius transformations on

• orbit of are the Ford circles

• point of tangency with , diameter =
• hyperbolic midpoint of the arc joining to this Ford circle is at euclidean height

• point of tangency with , diameter =
• hyperbolic midpoint of the arc joining to this Ford circle is at euclidean height

Proposition

• arc joining has -length
• -length = length of the portion outside Ford circles tangent to the real line at its endpoints

transitive on ,

• can suppose and
• joins Ford circle tangent at and another of diameter
• hyperbolic length of portion outside these is

pairing arcs and curves

• modular torus obtained from a pair of ideal triangles by identification
• blue arc is the unique arc disjoint from blue curve

Lemma A

The -length of the unique arc disjoint from the
simple closed geodesic such that is .

Proof: Easy calculation

Corollary B

Every Markoff number is a sum of squares ie

Proof of corollary

• simple close geodesic is invariant under the elliptic involution
• the unique arc disjoint from is invariant
• a fixed point of the elliptic involution on
• a lift of which is a vertical line and which meets
• since -length of = m this point is at euclidean height

and so we have the equation

by the same argument....

Lemma C

Let be a positive integer.
The "number of ways" of writing as a sum of squares

with coprime integers is equal to the number of arcs

coprime to which meet

Counting solutions

The "number of ways" of writing as

• eight solutions
• four choices for the signs
• swap and
• only swapping "counts"

Example

Lemma C

The number of ways of writing as a sum of squares

is equal to the number of arcs on the modular surface

1. of -length
2. which pass through the cone point of of order 2.

exactly 6 simple arcs of -length on

Proof of Button

every Markoff number is the sum of two squares

• if is prime then there are two ways of doing this
• there are oriented simple arcs of length on
• the multiplicity of in the Markoff tree is at most 6

In fact....

if m is a Markoff number and

• or

then m satisfies the uniqueness conjecture

Theorem (Fermat)

Let be a prime then has a solution over

• iff or is a multiple of 4.