principal congruence links for discriminant d=-3
TRANSCRIPT
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Principal Congruence Links for Discriminant D = −3
Matthias Goerner
UC Berkeley
April 20th, 2011
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Overview
Thurston congruence link, geometric description
Bianchi orbifolds, congruence and principal congruence manifolds
Results implying there are finitely many principal congruence links
Overview for the case of discriminant D = −3
Preliminaries for the construction
Construction of two more examples
Open questions
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Thurston congruence link
M−32+ζ
Complement is non-compact finite-volume hyperbolic 3-manifold.
Tesselated by 28 regular ideal hyperbolic tetrahedra.
Tesselation is “regular”, i.e., symmetry group takes every tetrahedronto every other tetrahedron in all possible 12 orientations.
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Cusped hyperbolic 3-manifolds
Ideal hyperbolic tetrahedron does not include the vertices.
Remove a small hororball. Ideal tetrahedron is topologically atruncated tetrahedron.
Cut is a triangle with a Euclidean structure from hororsphere.
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Cusped hyperbolic 3-manifolds have toroidal ends
Truncated tetrahedra form interior of a 3-manifold M̄ with boundary.
∂M̄ triangulated by the Euclidean triangles.
∂M̄ is a torus.
Ends (cusps) of hyperbolic manifold modeled on torus × interval.
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Knot complements can be cusped hyperbolic 3-manifolds
Cusp homeomorphic to a tubular neighborhood of a knot/linkcomponent.
Figure-8 knot complement tesselated by two regular ideal tetrahedra.
Hyperbolic metric near knot so dense that light never reaches knot.
Complement still has finite volume.
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“Regular tesselations”
(Source: wikipedia)
Spherical 2-dimensional version of “regular tesselations”: Platonicsolids.
Person in a tile cannot tell through intrinsic measurements in whattile he or she is or at what edge he or she is looking at.
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Two more examples
M−33
54 regular ideal tetrahedra
+1 +1
M−32+2ζ
120 regular ideal tetrahedra
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Thurston congruence link and the Klein quartic
xy3 + yz3 + zx3 = 0
Faces of ideal tetrahedra form immersed hyperbolic surface.
Filling the punctures yields an algebraic curve in CP2: Klein quartic.
Orientation-preserving symmetry group of the hyperbolic surface:PSL(2, 7), the unique finite simple group of order 168.
Thurston/Agol, “Thurston congruence link”
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Bianchi orbifolds
OD : ring of integers in Q(√D). D < 0,D ≡ 0, 1(4) discriminant.
Bianchi group:
PGL(2,OD) respectively PSL(2,OD)
is a discrete subgroup of PGL(2,C) ∼= PSL(2,C) ∼= Isom+(H3).
Bianchi orbifold:
MD1 =
H3
PGL(2,OD)respectively
H3
PSL(2,OD).
Every cusped arithmetic hyperbolic manifold is commensurable with aBianchi orbifold.
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Bianchi orbifolds
63
3
2
2
2
M−31
regular ideal tetrahedrondivided by
orientation-preservingsymmetries
44
3
2
2
2
M−41
regular ideal octahedrondivided by
orientation-preservingsymmetries
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Congruence subgroups
Fix ideal I in OD .
OD → OD/I induces map
p : PGL(2,OD)→ PGL(2,OD/I )
Congruence subgroup:
p−1(G ) for some subgroup G ⊂ PGL(2,OD/I ).
Principal congruence subgroup:
ker(p) = p−1(0).
(Principal) congruence manifold/orbifold: quotient of H3
MDz =
H3
ker(PGL (2,OD)→ PGL
(2, OD〈z〉
))Thurston congruence link complement is M−32+ζ .
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Cuspidal Cohomology, Baker’s links
Cuspidal cohomology yields an obstruction:If MD
1 can be covered by a link complement, then D ∈ L where L =
{−3,−4,−7,−8,−11,−15,−19,−20,−23,−24,−31,−39,−47,−71}.
Mark Baker constructed “some” cover for each D ∈ L, making it ‘iff’.
His covers are neither canonical nor regular.
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Finitely many principal congruence links
Gromov and Thurston 2π-Theorem: Dehn filling cusps of ahyperbolic manifold along peripheral curves with length > 2π yieldshyperbolic manifold again.(Length measured on embedded hororballs)
Agol and Lackenby: improved bound to > 6.
Corollary: If the shortest curve on every cusp has length > 6, themanifold is not a link complement.
Hence, only finitely many principal congruence manifolds MDz are link
complements.
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The case of discriminant D = −3
0 1 2 5 6 ≡ 2(1 + ζ)2
1 + ζ 2 + ζ
3 + 3ζ ≡ (1 + ζ)3
orbifolds within this circle
PGL not solvableif z prime and outside this circle
Gromov/Thurston’s bound
Agol/Lackenby’s bound
RP3
found link/orbifold diagram in S3
proved that not a link in S3
showed that link in RP3
RP3
3 ≡ (1 + ζ)2
2 + 2ζ = 2(1 + ζ)
4 = 22
4 + ζ ≡ (1 + ζ̄)(2 + ζ̄)
4 + 2ζ = 2(2 + ζ)
H3
ker(PGL (2,Z[ζ])→ PGL
(2, Z[ζ]〈z〉
))with ζ = e2πi/3
M−3z =H3
ker(PGL (2,Z[ζ])→ PGL
(2, Z[ζ]〈z〉
))with ζ = e2πi/3
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The case of discriminant D = −3
?
?0 1 2 5 6 ≡ 2(1 + ζ)2
1 + ζ 2 + ζ
3 + 3ζ ≡ (1 + ζ)3
orbifolds within this circle
PGL not solvableif z prime and outside this circle
Gromov/Thurston’s bound
Agol/Lackenby’s bound
RP3
found link/orbifold diagram in S3
proved that not a link in S3
showed that link in RP3
3 ≡ (1 + ζ)2
2 + 2ζ = 2(1 + ζ)
4 = 22
4 + ζ ≡ (1 + ζ̄)(2 + ζ̄)
4 + 2ζ = 2(2 + ζ)
H3
ker(PGL (2,Z[ζ])→ PGL
(2, Z[ζ]〈z〉
))with ζ = e2πi/3
N−3z =H3
ker(PSL (2,Z[ζ])→ PSL
(2, Z[ζ]〈z〉
))with ζ = e2πi/3
different from PGL
same as PGL
link
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The case of discriminant D = −3
?
?0 1 2 5 6 ≡ 2(1 + ζ)2
1 + ζ 2 + ζ
3 + 3ζ ≡ (1 + ζ)3
Gromov/Thurston’s bound
Agol/Lackenby’s bound
RP3
found link/orbifold diagram in S3
proved that not a link in S3
showed that link in RP3
3 ≡ (1 + ζ)2
2 + 2ζ = 2(1 + ζ)
4 = 22
4 + ζ ≡ (1 + ζ̄)(2 + ζ̄)
4 + 2ζ = 2(2 + ζ)
z-universal regular cover N̂−3z
with ζ = e2πi/3
different from PSL
same as PSL
link
orbifolds within this circle
infinite
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Preliminaries: Orbifolds
3-orbifold M locally modeled on quotient
R3
Γ→ U ⊂ M
by a finite subgroup Γ ⊂ SO(3,R).
Here, 3-orbifolds M are oriented.
Underlying topological space X (M) is a 3-manifold.
Singular locus Σ(M) is the set where Γ is non-trivial.Σ(M) is embedded trivalent graph with labeled edges.
Near edges of Σ(M): modeled on branched cover, Γ cyclic.
Near vertices of Σ(M): Γ is dihedral or orientation-preservingsymmetries of a Platonic solid.
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Orbifold notation
25
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Cusp(remove knot)
Edge of singular graph(modeled on branched cover)
Vertex of singular graph(modeled on orientation-preserving trian-gle group)
Cusp of orbifold(remove a small ball of underlying topo-logical manifold)
S1 ⊂ S3 consisting of ∞ and line perpen-dicular to paper plane
Surgery on knot
n right-handed full (360◦) twists
strands marked with arrow do notparticipate in twist but go under-neath
Figure 1.3: Conventions for orbifold diagrams.
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Construction of M−33
M−33 has 54 regular ideal tetrahedra and 12 cusps.
The orientation-preserving symmetries are PGL(
2, Z[ζ]〈3〉
)Lemma: M−33 → M−31+ζ is the universal abelian cover of M−31+ζ .
Lemma: The holonomy of this cover is given by
πorb1
(M−31+ζ
)�
(Z3
)3
.
Reason: 〈3〉 = 〈1 + ζ〉2 and Z[ζ]〈1+ζ〉
∼= Z/3.
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Overview of construction of M−33 27
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M̃II1+ζ M̃IV
1+ζ
M̃III1+ζM̃I
1+ζ
M−31+ζ
M−33
Figure 1.5: Abelian covers of the Bianchi orbifold for O−3 involved in the construction ofM−3
3 . Each arrow is a 3-cyclic cover.Matthias Goerner (UC Berkeley) Principal Congruence Links April 20th, 2011 21 / 44
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Step 1 of M−33 : 3-cyclic cover along unknot
28
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M̃I1+ζ
M−31+ζ
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Figure 1.6: Construction of M̃I1+ζ .
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Step 2 of M−33 : 3-cyclic cover of (3, 3, 3)-triangle orbifold
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M̃I1+ζ
M̃II1+ζ
Figure 1.7: Construction of M̃II1+ζ .
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Step 3 of M−33 : Divide out 3-cyclic symmetry
The singular locus is too complicated to construct a 3-cyclic cover.
Divide out 3-cyclic symmetry.
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Step 4 of M−33 : 3-cyclic cover of (3, 3, 3)-triangle orbifold
30
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M̃III1+ζ
M̃IV1+ζ
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Figure 1.8: Construction of M̃IV1+ζ .Matthias Goerner (UC Berkeley) Principal Congruence Links April 20th, 2011 25 / 44
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Step 5 of M−33 : Cover according to Akbulut and Kirby
Akbulut and Kirby, “Branched Covers of Surfaces in 4-Manifolds”:Construction of cyclic cover of B4 branched over Seifert surface of alink in S3 = ∂B4 pushed into B4.
Here, we are only interested in what happens on the boundary S3.
The Seifert surface will determine the holonomy of the cyclic coverbranched over a link in S3.
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Example of a cyclic cover
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Example of a cyclic cover
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Example of a cyclic cover
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Example of a cyclic cover
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Step 5 of M−33 : Cover according to Akubulut and Kirby
32
Figure 1.10: Construction of 3-cyclic branched cover M−33 → M̃IV
1+ζ . The resulting linkrepresentation of M−3
3 has four +2 surgeries.
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Rolfsen twists
(Source: Rolfsen, Knots and Links)
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Step 5 of M−33 : Rolfsen twists and blow-downs
35
Figure 1.13: The link for M−33 .
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Dihedral symmetry of link for M−33
M−33
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Construction of M−32+2ζ
M−32+2ζ has 120 regular ideal tetrahedra and 20 cusps.
Orientation-preserving symmetries are
PGL
(2,
Z[ζ]
〈2 + 2ζ〉)
)∼= PGL
(2,
Z[ζ]
〈1 + ζ〉
)⊕ PGL
(2,
Z[ζ]
〈2〉
)∼= S4 ⊕ A5.
For G ⊂ S4 ⊕ A5, let
|G | =M−32+2ζ
G.
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Construction of M−32+2ζ
Orbifold M−32 and manifold double-cover in:Dunfield, Thurston, “The virtual Haken conjecture: experiments andexamples”
Decktransformation group of
M−32+2ζ∼= |0| → |S4 ⊕ 0| ∼= M−32
is S4, a solvable group.
S4 and Z/5 ⊂ A5 commute in S4 ⊕ A5.Can divide 5-cyclic symmetry and postpone 5-cyclic cover until later.
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Overview of the construction of M−32+2ζ
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M−31
∼=∣∣∣PGL
(2, Z[ζ]
2+2ζ
)∣∣∣ ∼= |S4 ⊕ A5|
|A4 ⊕ Z/5|M−32
∼= |S4 ⊕ 0| M−31+ζ
∼= |0⊕ A5|
M−32+2ζ
∼= |0|
|A4 ⊕ 0|∣∣∣(Z/2)2 ⊕ Z/5
∣∣∣
|Z/2⊕ Z/5|
S4A5
Z2 ⊕ A5
Z3
Z2
Z10
Z2
S4 A5
Z5
Figure 1.14: Covers of the Bianchi orbifold for O−3 involved in the construction of M−32+2ζ .
An arrow indicates a regular cover if and only if it is labeled by the group of Decktransfor-mations.
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Pentacle
Pentacle orbifold =Minimally twisted 5-component chain link
involution around dotted
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Tricks for M−32+2ζ
Blow-up makes 5-cyclic symmetry of chain link visible.
Rolfsen twists produce surgery unknots with coefficients ab with p|b.
These unknots serve as branching locus for Akbulut and Kirbyconstruction.
Reduce rational plumbing diagrams to single surgery unknot revealinglens space structure.
Projection onto torus for visualization.
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M−32+2ζ in RP3
-2
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M−32+2ζ in S3
+1 +1
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Progress on the missing links
z = 3 + ζ, 3 + 2ζ, 5 + ζ is prime.For z = 3 + ζ:
Let G =
{(1 x0 1
)}.
Triangulation of M = H3/p−1(G ) (Python script).
M−33+ζ is unique (as manifold) 13-cyclic cover of M with 14 cusps.
M obtained by 143 Dehn filling of 10365.
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Open questions
Find remaining 5 potential principal congruence links, or showmanifolds are not link complements.
Is PGL or PSL more natural?
Are there infinitely many congruence links?
Are there infinitely many regular Bianchi orbifold cover links?
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Classification of regular Binachi orbifold covers for D = −3
Invariant of regular Bianchi orbifold cover: Cusp shape z .Triangulation by regular tetrahedra induces lattice Z[ζ] ⊂ C on cusps.Cusp torus is C/〈z〉 for some z ∈ Z[ζ] determined up to unit.
Fix z . Category of regular Bianchi orbifold covers:
Finite-volume initial object for
z ∈ {2, 2 + ζ, 2 + 2ζ, 3, 3 + ζ, 3 + 2ζ, 4, 4 + ζ}.
Terminal object is M−3z for
z ∈ {2 + ζ, 3 + ζ}.
For the lower z , we have already seen all regular Bianchi orbifold covers.
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