Hyperbolic geometry and Diophantine approximation

greg mc shane

Contents

  • prelude
  • numbers: integers, rationals, quadratic irrationals
  • geodesics on the modular torus and identity
  • Dehn twists and Markoff numbers
  • two index 6 subgroups
  • three punctured sphere, automorphisms
  • modular torus, automorphisms

Simple geodesics

  • A curve covers a simple curve in
    if and only
    is a (primitive) parabolic element of fixing

  • Definition
    The cusp region (of area ) is the quotient of
    the horoball
    by the subgroup

  • Lemma A
    If projects to a simple closed curve in then it is either a vertical line or disjoint from the horoball with .

  • Lemma A If projects to a simple closed geodesic in then it is either:
    • a vertical line or
    • disjoint from the horoball with (in grey).

Part 1

Numbers

  • Boris Springborn, The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms. Enseign. Math.,
  • rationals
    • simple closed curves on the torus
  • irrationals
    • quadratic irrationals eg golden ratio
    • other irrationals

Continued fraction of

where and for .

  • we write
    • golden ratio

Best approximation of by rationals

Truncating the continued fraction of at the -th step gives a rational approximation the n-th convergent of .

  • This is the best approximation of by rationals with
    denominator at most .
  • For example, the best approximations of are

  • where is the -th Fibonacci number.

Visualization

  • Farey diagram:
    • vertices = rationals, edges = Farey neighbors
    • are Farey neighbors iff
  • invariant translation of

ctd frac

Visualization golden ratio = 1.6180339...

ctd frac

Visualization golden ratio = 1.6180339...

ctd frac

ctd frac

Constant Continued Fraction Notation First 5 Convergents (Fractions)
Golden Ratio ()
Square Root of 2 ()
Euler's Number ()
Pi ()
Cube Root of 3 ()

Can "read" the from the picture:

  • the number of corners of the "zig-zag" alternately on the left and the right of the vertical line ending at
  • The modular surface and continued fractions, Caroline Series

Visualization golden ratio = 1.6180339...

ctd frac

zig-zag

Best approximations alternate on the left and the right

  • rational numbers are poorly approximated by rationals

continued fraction is finite

  • quadratic irrationals are also poorly approximated by rationals

continued fraction is (pre)periodic, eg

Irrational Number (x) Constant Cx Numerical Value
Golden Ratio (φ) 1/ 0.4472
1/ 0.4472
1/(2 0.3535
1/(2 0.2886
  • Note: This is related to how deep a (closed) geodesic can go into the cusp of the modular surface.

Part 2

email

  • My name is Jia Longsong, and I am a Ph.D. student at Peking University. Recently, I have been studying your remarkable identity in detail. In your Acknowledgements, you mentioned that the motivation for the work lies in two series of seminars held at the University of Warwick. The first series concerned Thurston's unpublished work on minimal stretch maps between surfaces and the second Maskit's embedding of the Teichmuller space of the punctured torus. I am truly fascinated by the profound ideas behind your work.

Minimal stretch maps

https://arxiv.org/abs/math/9801039

  • This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps (minimal stretch maps) and geodesics for the `Lipschitz metric' are constructed. The extremal Lipschitz constant equals the maximum ratio of lengths of measured laminations, which is attained with probability one on a simple closed curve.

  • Geodesic laminations = limits of simple closed curves

  • Definition The Birman-Series set (BS) is the union of the pointset of all compact measured laminations on a hyperbolic surface.
    In fact this is just the closure of the set of the union of all
    simple closed geodesics.

  • Theorem 10.2 (Stretch maps)
    For any hyperbolic surface of finite area,
    the Hausdorff dimension of BS is 0.
    In particular it is:

    • nowhere dense
    • has measure zero.

Questions

  • nowhere dense, so where are the holes?

  • has measure zero, can we exploit this?

  • Answer for a punctured torus:

Punctured torus

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

  • Definition
    The (Haas)cusp region is the quotient of the
    horoball by

  • Lemma A+
    If is a simple closed geodesic
    then it is disjoint from the Haas cusp region.

  • Lemma A
    If projects to a simple curve in then it is either

    • a vertical line (lift of arc )
    • or disjoint from the horoball with (lift of )

  • Corollary
    The cusp region is a hole in the Birman-Series set.

  • The extended Birman-Series set (EBS) is the union of the pointset of all measured laminations on a hyperbolic surface.
    In fact this is just the closure of the set
    of the union of all simple closed geodesics and arcs.

  • Theorem 10.2 (essential stretch maps)
    For any hyperbolic surface of finite area,
    the Hausdorff dimension of EBS is 0.
    In particular it is

    • nowhere dense
    • and has measure zero.

nowhere dense, so where are the holes in the cusp region?

Geodesics in homology classes

  • Definition: Let be an essential closed curve
    on the punctured torus and denote its length.

  • Natural map:

  • Theorem
    The shortest representative for a non trivial homology class is always a multiple of a closed simple geodesic.
    If are coprime then it is a single geodesic .

Dehn twist round acting on arcs

  • Dehn twist
  • Dehn twist

limit geodesic on pants

Previous slide is the infinite cyclic cover of a pair of pants
obtained by cutting the torus along the geodesic .

limit geodesic on torus

  • Pants glued up to a torus
  • Shaded region is a pair of gaps in EBS

torus cut open along lamination

  • Theorem A
    If is an arc then it is isolated in the
    It sits in the center of a pair of gaps of area

  • Corollary

Dehn twist round acting on arcs

Dehn twist

The sequence of arcs converges to a geodesic that is

  • disjoint from the lifts of the arc
  • asymptotic to a pair of lifts of the closed geodesic .

Dehn twist acting on arcs

  • grey region is gap in EBS
  • dotted red lines are lifts of arcs

  • To promote this to an equality we need to show :
    1. show every gap is "adjacent" to an arc
    2. then apply Theorem 10.2 to show that the union of the gaps has
      full measure in the cusp region.

Show every gap is "adjacent" to an arc:

  • extend the segment bounding a gap to a complete geodesic
  • look at it in the homology cover, it's essentially a line of slope .
  • if is irrational use the continued fraction expansion of to find an infinite sequence of arcs converging to this line alternately from left and right. contradiction
  • so is rational and the geodesic is asymptotic to the lift of a
    simple closed geodesic.

Part 3

Another problem

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 Markoff triple
determines the two other numbers.

Uniqueness conjecture: tree structure

  • 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

Partial results

m = Markoff number then there are at most 6 triples containing m.

  • Odd index Fibonacci numbers odd index Pell numbers =


source

Visualizing using natural map

Markoff numbers

  • action on
  • transitive mapping class group action on essential simple curves on

Tree structure

comes from Bass-Serre tree of

Role of the character variety

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

  • Definition character map

  • Theorem: (Fricke, Cohn and many others) the semi-algebraic set:

  • can be identified with the Teichmueller space
    of the punctured torus.
  • there is a finite index subgroup of the automorphisms induced by the action of the mapping class group
    • the permutation
    • the involution (the flip)

pairing arcs and curves

simple closed curve
coprime integer

  • simple closed geodesic
  • hyperbolic length
  • arc disjoint from the closed geodesic
  • -length of

traces= lengths

  • Lemma B
    For an appropriate normalization:

-length of

  • appropriate normalization
    = choice of cusp region

Proof (by recurrence):

  • Let be a pair of simple closed geodesics
    on the punctured torus that intersect once.
  • The corresponding traces together
    with some other z>0$ satisfythe Markoff equation:

  • Call the solutions . These satisfy Vieta's relations:
    1. (Trace relation in )
    2. (Ptomley relation for -lengths)

  • Bugeaud, Reutenauer, Siksek; A Sturmian sequence related to the uniqueness conjecture for Markoff numbers:
    Odd index Fibonacci Odd index Pell =


  • Up to a multiplicative factor these are sequence of
    traces of simple closed geodesics ( lengths of arcs)
    obtained by doing Dehn twists on the modular torus.

  • Traces

  • Theorem
    Let be a simple closed geodesic on the punctured torus and the unique arc disjoint from . and a pair of arcs such that are the sides of an embedded ideal triangle.
    Then
    1.the (normalised) -lengths of is a Markoff triple.

    1. any power of Dehn twist along gives a new Markoff
      triple containing the -length of trace of .

Unicity conjecture

  • Then
    1.the (normalised) -lengths of is a Markoff triple.

    1. any power of Dehn twist along gives a new Markoff
      triple containing the -length of trace of .

  • Unicity conjecture is equivalent to the statement that
    no fractional Fenchel-Nielsen twist along
    gives integer -lengths to the images of .

Bugeaud, Reutenauer, Siksek

  • This is equivalent to a statement like:

  • no "fractional" twist along one of the
    takes the arc of length that meets
    to an arc of -length .

Bugeaud, Reutenauer, Siksek

How was I going to prove it?

  • is the intersection of
    the line and the axis of
  • use Dehn twists to construct a sequence of points "too quickly" for infinitely many to be rational.

Dehn twist of the axes

  • dotted red lines are lifts of arcs
  • is the axis of

Where did I go wrong? Reciprocal geodesics

  • is conjugate to it's own inverse by

  • the associated (simple) closed geodesic is invariant by the
    elliptic involution

  • any closed simple geodesic on the punctured torus is invariant
    under the elliptic involution

  • so I picked a lifts of a sequence of simple closed geodesics
    with traces that all went through the fixed point of ie .

Where did I go wrong? Reciprocal geodesics

  • any closed simple geodesic on the punctured torus is invariant
    under the elliptic involution
  • so I picked a lifts of a sequence of simple closed geodesics
    with traces that all went through the fixed point of .
  • These converge to the axis of and the intersection points give me the result
  • Unfortunately not

Dehn twist of the axes

  • dotted red lines are lifts of arcs
  • is the axis of

  • is the axis of
  • limit of the is intersection of with black semicircle
  • black semicircle projects to a leaf of the lamination containing
    = "limit" of the Dehn twists

THE END

#

<!-1- _transition: cube -1->

- slides : google **greg mcshane github**

- click on **serfest**

- if there is a bug in my slides blame [this guy](https://github.com/yhatt)

$\alpha \in H^1(T,\mathbb{Z}), \, \| \alpha \| := \inf_{ c \in \gamma} \ell_c/2$

#

## infinity of Markoff triples: $z=1$

$\begin{pmatrix} 3 & -1 \\ 1 & 0 \end{pmatrix}$

is an automorph of

$$x^2 + y^2 - 3x y.$$

So $( v_n,v_{n+1},1)$ is a Markoff triple where

$\begin{pmatrix} x \\ y \end{pmatrix}= \begin{pmatrix}v_{n+1} \\ v_n \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ 1 & 0 \end{pmatrix}^n \begin{pmatrix}1 \\ 1 \end{pmatrix}$

* Conclusion too hard!!!

![w:500px](./Markoff_tree_full.svg)

{: style="text-align: left"}

$$A^nBA^n = \begin{pmatrix} F_{2n-3}F_{2n-1} & F_{2n}^2 + F_{2n-2}F_{2n} \\ F_{2n}^2 - F_{2n-2}F_{2n} & F_{2n-1}F_{2n+1} + F_{2n}^2 \end{pmatrix}$$