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