Thin Groups and Fractals - WordPress.com · Introduction: Hyperbolic GeometryRulesTranslationsApollonian PackingK3’sFurther Work WCNT 2016 Thin Groups and Fractals Daniel Lautzenheiser

Introduction: Hyperbolic Geometry Rules Translations Apollonian Packing K3’s Further Work

WCNT 2016

Thin Groups and Fractals

Daniel Lautzenheiser and Arthur Baragar

Department of Mathematical SciencesUniversity of Nevada Las Vegas

December 14, 2016

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Spherical Geometry

S = {~x : ||~x || = 1} = {~x : ~x · ~x = 1} = {~x : ~x t I~x = 1}Lines: S ∩ PAngles: ~x · ~y = ||~x ||||~y || cos θWeighted dot product: I 7→ A, ( A positive-definite, symmetric)

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Lorentz Product

~x ◦ ~y = ~x tJ~y , J =

1 0 00 1 00 0 −1

” ◦ ” is bi-linear and symmetric, but not an inner product.

Light Cone:

L+ = {~x : ~x ◦ ~x = 0}

Outer Hyperboloid (1 sheet) :

~x ◦ ~x = 1

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Pseudosphere

Pseudosphere model:

V = {~x : ~x ◦ ~x = −1}

V+ = {~x = (x , y , z) ∈ V : z > 0}

Lines: P ∩ V+

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Disk Model

Stereographic Projection:

π : V+ −→ D

D = {(x , y , 0) : x2 + y2 < 1}

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Disk Model Cont.

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Disk Model: Lines

~n1 = (1, 0, 0)~n2 = (−1, 1, 0)~n3 = (1, 1, 1)Intersect

planes :

~x ◦ ~n1 = 0~x ◦ ~n2 = 0~x ◦ ~n3 = 0

with V+ to get lines. Apply π to theselines.

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Disk Model: Moving Points

The vectors ~n1, ~n2, and ~n3 define 3planes. The mapping

Rj (~x) = ~x − 2Proj~nj~x = ~x − 2

~x ◦ ~nj

~nj ◦ ~nj~nj

is reflection through the plane definedby ~nj . Rj is an isometry since

Rj (~x) ◦ Rj (~y) = ~x ◦ ~y

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Disk Model: Lattice Points

(8, 4, 9) 7→R2 (4, 8, 9) 7→R3 (−2, 2, 3) 7→R1 (2, 2, 3) 7→R3 (0, 0, 1)

I F is the fundamental domain forthe group Γ = {R1,R2,R3}

I Γ acts transitively on the latticepoints of V+

I F is a Coxeter polyhedronI Method of descent to F

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What does a picture look like after inversion in a circle?

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Process: Pictures to/from Lorentz Product

Rules:

1. Choose a basis: {e1, · · · , en}2. Construct J = [ei · ej ]...

Example:

I Think of ” · ” as ” ◦ ”

I Choose e1 · e1 = e2 · e2 = −2I Since e3 is the point at infinity,

e3 · e3 = 0I e1 · e2 = ±||e1||||e2|| cos θ or...

e1 · e2 = ±||e1||||e2|| coshψI e1 · e3 = curvature (1/radius)

J =

−2 d 1d −2 11 1 0

I J induces "◦"

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Fundamental Domain

{l1 · e1 = 0⇐⇒~l1 ◦ ~e1 = 0l1 · e3 = 0⇐⇒~l1 ◦ ~e1 = 0{

n1 · e1 = 0n1 · e2 = 0

{l3 · n1 = 0l3 · e3 = 0

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Fundamental Domain

~l1 = (1,−1, 2 + d), ~n1 = (1, 1, 2− d)

L1 =

1 2 00 −1 00 2d + 4 1

,N1 =

1 0 2d−2

0 1 2d−2

0 0 −1

, L3 =

0 1 01 0 00 0 1

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Hyperbolic Translations

L1L3 ∼

1 1 00 1 10 0 1

N1L1 ∼

d+6+4

√d+2

d−2 0 0

0 d+6+4√

d+2d−2 0

0 0 1

I Since N1L1 is an isometry,det(N1L1) = 1 = product ofeigenvalues.

I If d = 1, 3, 4 then we getfundamental units in the ring ofintegers Z[

√D] ⊆ Q[

√D], some

D ∈ N.I Example: If d = 4, then

Spec(N1L1L3) = {7± 4√

3,−1},Spec(N1(L1L3)2) ={25± 4

√39,−1}.

(Pell’s eqn. for D=3,39)

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Translations cont.Let Λ = {N1(L1L3)k}∞k=1. How big is Spec(Λ)?

Values of the Discriminant (Within Eigenvalue)

k d=1 d=3 d=41 6 30 32 33 105 393 78 230 214 141 5 35 222 70 576 321 905 3277 438 1230 1118 573 1605 5799 6 2030 18310 897 2505 903

Guess (based on first 10,000 cases): Spec(Λ) has units in infinitely many D,but misses some values e.g. D = 7.

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Ji = [ei ·ej ] =

−2 2 2 42 −2 2 42 2 −2 04 4 0 0

, Γ = 〈R1,R2,R3,R4,R5〉 ,T = 〈R1,R2,R3,R4〉

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Connection to K3 surfaces

Given a (class of) K3 surface, we may describe a hyperbolic cross section ofthe ample cone.

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I Each previous J is even (even entries on main diagonal), symmetric,and has signature (1, n − 1).

I By [Mor84] and the Hodge Index Theorem, there exists a class of K3surfaces with intersection matrix J = [ei · ej ].

I So, these fractals are ample cones of K3 surfaces.I "Apollonian K3 surfaces"

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Further Work

1. Can we modify or generalize the matrix J in the hyperbolic translationscase to hit more values of D?

2. Generalized method of descent in "non-Coxeter" cases

3. Using kJ−1 as a Lorentz product, some k ∈ N.

"Dual" belt packing:

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References

[1] Stange, Katherine. The Apollonian Structure of Bianchi Groups.Department of Mathematics, University of Colorado. 2015[2] Dolgachev, Igor. Orbital Counting on Algebraic Surfaces and SpherePackings. Springer International Publishing Switzerland 2016.[3] Baragar, Arthur. Automorphisms on K3 Surfaces with Small PicardNumber. Department of mathematical sciences, UNLV. 2014[4] Morrison, D. R. "On K3 Surfaces with Large Picard Number." InventionesMathematicae Invent Math 75.1 (1984): 105-21. Web.[5] Boyd, David W. A New Class of Infinite Sphere Packings. Pacific Journalof Mathematics. 1974[6] Baragar, Arthur. The Neron-Tate Pairing and Elliptic K3 Surfaces.Preprint. 2016[7] Baragar, Arthur. Lattice Points on Hyperboloids of One Sheet. New YorkJournal of Mathematics. 2013.

Special thanks to Sarah Glaser for helping me with some of the graphics

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Connections to K3 surfaces

I K3 surfaces are examples of elliptic fibrations There are smooth curves(divisor classes) passing through each elliptic curve.

I The isogeny P 7→ −P sends one smooth curve to another and inducesan automorphism of the K3 surface.

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Ample Cone Symmetries

Ample cone symmetries←→arithmetic on elliptic curves

I O 7→ O + P (horizontaltranslation fixing∞)

I O 7→ O + Q (diagonal translation)I P 7→ −P (−1 map through O)

Jamp =

−2 2 2 42 −2 2 42 2 −2 44 4 4 0

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Thin Groups and Fractals - WordPress.com · Introduction: Hyperbolic GeometryRulesTranslationsApollonian PackingK3’sFurther Work WCNT 2016 Thin Groups and Fractals Daniel Lautzenheiser