Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Geometric structures on the Figure EightKnot Complement
ICERM WorkshopExotic Geometric Structures
Martin Deraux
Institut Fourier - Grenoble
Sep 16, 2013
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
The figure eight knot
Various pictures of 41:
K = figure eight knot
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
The complete (real) hyperbolic structure
M = S3 \ K carries a complete hyperbolic metric
M can be realized as a quotient
Γ \ H3R
where Γ ⊂ PSL2(C) is a lattice (discrete group with quotientof finite volume)
I One cusp with cross-section a torus.
I Discovered by R. Riley (1974)
I Part of a much more general statement about knotcomplements/3-manifolds (Thurston)
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Holonomy representation
For example by Wirtinger, get
π1(M) = 〈g1, g2, g3 | g1g2 = g2g3, g2 = [g3, g−11 ] 〉,
with fundamental group of the boundary torus generated by
g1 and [g−13 , g1][g−1
1 , g3]
Alternatively
π1(M) = 〈a, b, t | tat−1 = aba, tbt−1 = ab 〉.
The figure eight knot complement fibers over the circle, withpunctured torus fiber.
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Holonomy representation (cont.)
Search for ρ : π1(M)→ PSL2(C) with ρ(g1) = G1,ρ(g3) = G3,
G1 =
(1 10 1
), G3 =
(1 0−a 1
)Requiring
G1[G3,G−11 ] = [G3,G
−11 ]G3
get a2 + a + 1 = 0, so
a =−1± i
√3
2= ω or ω.
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Ford domain for the image of ρ
Bounded by unit spheres centered in Z[ω], ω = −1+i√
32
Cusp group generated by translations by 1 and 2i√
3.
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Prism picture
x2
x3 x3 x3
x3x3
x4
x4
x4x2
x1 x1
x1x1
x4
x4
x3 x2
x4
x2 x1
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Triangulation pictureCan also get the hyperbolic structure gluing two idealtetrahedra, with invariants z , w .
Compatibility equations:
z(z − 1)w(w − 1) = 1
For a complete structure, ask the boundary holonomy tohave derivative 1, and this gives
z = w = ω
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Complete spherical CR structures
Spherical CR structure arising as the boundary of a ballquotient.
Ball quotient: Γ \ B2, where Γ is a discrete subgroup ofBihol(B2) = PU(2, 1).
The manifold at infinity inherits a natural spherical CRstructure, called “complete” or “uniformizable”.
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Domain of discontinuity
Γ ⊂ PU(2, 1) discrete
I Domain of discontinuity ΩΓ
I Limit set ΛΓ = S3 − ΩΓ
The orbifold/manifold at infinity of Γ is Γ \ ΩΓ
I Manifold only if no fixed points in ΩΓ (isolated fixedpoints inside B2 are OK);
I Can be empty (e.g. when Γ is (non-elementary) and anormal subgroup in a lattice).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Biholomorphisms of B2
Up to scaling, B2 carries a unique metric invariant under thePU(2, 1)-action, the Bergman metric.
B2 ⊂ C2 ⊂ P2C
With this metric: complex hyperbolic plane.
I Biholomorphisms of B2: restrictions of projectivetransformations (i.e. linear tsf of C3).
I A ∈ GL3(C) preserves B2 if and only if
A∗HA = H
where
H =
−1 0 00 1 00 0 1
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Equivalent Hermitian form:
J =
0 0 10 1 01 0 0
Siegel half space:
2Im(w1) + |w2|2 < 0
Boundary at infinity ∂∞H2C (minus a point) should beviewed as the Heisenberg group, C× R with group law
(z , t) ∗ (w , s) = (z + w , t + s + 2Im(zw)).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
I Copies of H1C (affine planes in C2) have curvature −1
I Copies of H2R (R2 ⊂ C2) have curvature −1/4 (linear
only when through the origin)
I No totally geodesic embedding of H3R!
For this normalization, we have
cosh1
2d(z ,w) =
|〈Z ,W 〉|√〈Z ,Z 〉〈W ,W 〉
where
I Z are homogeneous coordinates for z
I W are homogeneous coordinates for w
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Isometries of H2C
Classification of (non-trivial) isometriesI Elliptic (∃ fixed point inside)
I regular elliptic (three distinct eigenvalues)I complex reflections
I in linesI in points
I Parabolic (precisely one fixed point in ∂H2C)
I Unipotent (some representative has 1 as its onlyeigenvalue)
I Screw parabolic
I Loxodromic (precisely two fixed points in ∂H2C)
PU(2, 1) has index 2 in IsomH2C (complex conjugation).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Central question
Which 3-manifolds admit a complete spherical CRstructure?
In other words:Which 3-manifolds occur as the manifold at infinity Γ \ΩΓ ofsome discrete subgroup Γ ⊂ PU(2, 1)?
Silly example: lens spaces! Take Γ generated by1 0 0
0 e2πi/q 0
0 0 e2πip/q
with p, q relatively prime integers (in this case ΩΓ = S3).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Central question
Which 3-manifolds admit a complete spherical CRstructure?
In other words:Which 3-manifolds occur as the manifold at infinity Γ \ΩΓ ofsome discrete subgroup Γ ⊂ PU(2, 1)?
Silly example: lens spaces! Take Γ generated by1 0 0
0 e2πi/q 0
0 0 e2πip/q
with p, q relatively prime integers (in this case ΩΓ = S3).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
More complicated examples
I Nil manifolds
I Lots of circle bundles (Anan’in-Gusevski, Falbel,Parker,. . . . . . )
(open hyperbolic manifolds)
I Whitehead link complement (Schwartz, 2001)
I Figure eight knot complement (D-Falbel, 2013)
I Whitehead link complement (Parker-Will, 201k , k ≥ 3)
(closed hyperbolic manifolds)
I ∞ many closed hyperbolic manifolds (Schwartz, 2007)
I ∞ many surgeries of the figure eight knot (D, 201k ,k ≥ 3).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
More complicated examples
I Nil manifolds
I Lots of circle bundles (Anan’in-Gusevski, Falbel,Parker,. . . . . . )
(open hyperbolic manifolds)
I Whitehead link complement (Schwartz, 2001)
I Figure eight knot complement (D-Falbel, 2013)
I Whitehead link complement (Parker-Will, 201k , k ≥ 3)
(closed hyperbolic manifolds)
I ∞ many closed hyperbolic manifolds (Schwartz, 2007)
I ∞ many surgeries of the figure eight knot (D, 201k ,k ≥ 3).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
More complicated examples
I Nil manifolds
I Lots of circle bundles (Anan’in-Gusevski, Falbel,Parker,. . . . . . )
(open hyperbolic manifolds)
I Whitehead link complement (Schwartz, 2001)
I Figure eight knot complement (D-Falbel, 2013)
I Whitehead link complement (Parker-Will, 201k , k ≥ 3)
(closed hyperbolic manifolds)
I ∞ many closed hyperbolic manifolds (Schwartz, 2007)
I ∞ many surgeries of the figure eight knot (D, 201k ,k ≥ 3).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Negative result
(not all manifolds)
I Goldman (1983) classifies T 2-bundles with spherical CRstructures.
For instance T 3 admits no spherical CR structure at all(complete or not!)
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
CR surgery (Schwartz 2007)
M a complete spherical CR structure on an open manifoldwith torus boundary.
If we have
1. The holonomy representation deforms
2. Γ \Ω is the union of a compact region and a “horotube”
3. Limit set is porous
Then ∞ many Dehn fillings Mp/q admit a completespherical CR structures.
Not effective, which p/q work?
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
The figure eight knot
Theorem(D-Falbel, 2013) The figure eight knot complement admits acomplete spherical CR structure.
Relies heavily on:
Theorem(Falbel 2008) Up to PU(2, 1)-conjugacy, there are preciselythree boundary unipotent representationsρ1, ρ2, ρ3 : π1(M)→ PU(2, 1).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
The figure eight knot
Theorem(D-Falbel, 2013) The figure eight knot complement admits acomplete spherical CR structure.
Relies heavily on:
Theorem(Falbel 2008) Up to PU(2, 1)-conjugacy, there are preciselythree boundary unipotent representationsρ1, ρ2, ρ3 : π1(M)→ PU(2, 1).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Preliminary analysis
Falbel 2008:
I Im(ρ1) C PU(2, 1,Z[ω])
I Im(ρ2) ⊂ PU(2, 1,Z[√−7])
In particular, ρ1 and ρ2 are discrete.
Im(ρ1) has empty domain of discontinuity (same limit set asthe lattice PU(2, 1,Z[ω])).
Action of Out(π1(M)) up to conjugation,
ρ3 = ρ2 τ
for some outer automorphism τ of π1(M) (orientationreversing homeo of M).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Preliminary analysis
Falbel 2008:
I Im(ρ1) C PU(2, 1,Z[ω])
I Im(ρ2) ⊂ PU(2, 1,Z[√−7])
In particular, ρ1 and ρ2 are discrete.
Im(ρ1) has empty domain of discontinuity (same limit set asthe lattice PU(2, 1,Z[ω])).
Action of Out(π1(M)) up to conjugation,
ρ3 = ρ2 τ
for some outer automorphism τ of π1(M) (orientationreversing homeo of M).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Complete structure for 41.
Write
I Γ = Im(ρ2),
I Gk = ρ2(gk).
G1 =
1 1 −1+√
(7)i
20 1 −10 0 1
G2 =
2 3−i√
72 −1
−3+i√
72 −1 0−1 0 0
G3 = G−1
2 G1G2.
I G1, G3 unipotent
I G2 regular elliptic of order 4.
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Back to Dirichlet domains
To see that ρ2 does the job, one way is to studyDirichlet/Ford domains for Γ2 = Imρ2.
DΓ,p0 =
z ∈ B2 : d(z , p0) ≤ d(γz , p0) ∀γ ∈ Γ
We will assume DΓ,p0 has non empty interior (hard to prove!)
Key:
1. When no nontrivial element of Γ fixes p0, this gives afundamental domain for the action of Γ.
2. Otherwise, get a fundamental domain for a cosetdecomposition (cosets of StabΓp0 in Γ).
More subtle:
1. Beware these often have infinitely many faces (Phillips,Goldman-Parker)
2. Depend heavily on the center p0.
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Back to Dirichlet domains
To see that ρ2 does the job, one way is to studyDirichlet/Ford domains for Γ2 = Imρ2.
DΓ,p0 =
z ∈ B2 : d(z , p0) ≤ d(γz , p0) ∀γ ∈ Γ
We will assume DΓ,p0 has non empty interior (hard to prove!)
Key:
1. When no nontrivial element of Γ fixes p0, this gives afundamental domain for the action of Γ.
2. Otherwise, get a fundamental domain for a cosetdecomposition (cosets of StabΓp0 in Γ).
More subtle:
1. Beware these often have infinitely many faces (Phillips,Goldman-Parker)
2. Depend heavily on the center p0.
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Back to Dirichlet domains
To see that ρ2 does the job, one way is to studyDirichlet/Ford domains for Γ2 = Imρ2.
DΓ,p0 =
z ∈ B2 : d(z , p0) ≤ d(γz , p0) ∀γ ∈ Γ
We will assume DΓ,p0 has non empty interior (hard to prove!)
Key:
1. When no nontrivial element of Γ fixes p0, this gives afundamental domain for the action of Γ.
2. Otherwise, get a fundamental domain for a cosetdecomposition (cosets of StabΓp0 in Γ).
More subtle:
1. Beware these often have infinitely many faces (Phillips,Goldman-Parker)
2. Depend heavily on the center p0.
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Poincare polyhedron theorem
Important tool for proving that DΓ,p0 has non-empty interior.
Use DF ,p0 instead of DΓ,p0 for some subset F ⊂ Γ.[In simplest situations, F is finite].
Assume
I F generates Γ;
I F = F−1 and opposite faces are isometric;
I Cycle conditions on ridges (faces of codimension 2)
I Cycles of infinite vertices are parabolic
Then the group Γ is discrete and we get
I An explicit group presentation 〈F |R〉 where R are cyclerelations.
I A list of orbits of fixed points.
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Dirichlet/Ford domains
Two natural choices for the center:
I Fixed point of G1 (unipotent) Ford domain
I Fixed point of G2 (regular elliptic) Dirichlet domain
Dirichlet: ∂∞DΓ is (topologically!) a solid torus.
Ford: ∂∞DΓ is a pinched solid torus (horotube).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Combinatorial structure of ∂(∂∞DΓ) for theDirichlet domain domain
5
p1 p4 p3 p2
q2q1
p2 p1 p4 p3 p2
q2 q3 q4
4 6 8
5317
1 3 5 7
287 1
4 628
64
2
p2
G2
3
q1
1 : G1p0
2 : G−11 p0
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Combinatorial structure of ∂(∂∞DΓ) for the Forddomain
1 G−11 (1)
G−11 (3)
G−11 (2)
G−11 (4)
G1(1)
G1(2)
G1(4)
G1(3)
G−31 (4)G−2
1 (4)
G31(1) G2
1(1)
3
2
4
1 : G2p0 2 : G−12 p0
3 : G3p0 4 : G−13 p0
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Computational techniques
The key is to be able to prove the combinatorics ofDirichlet/Ford polyhedra.
Need to solve a system of quadratic inequalities in fourvariables.
I Guess faces and incidence by numerical computations(grids in parameters for intersections of two bisectors)
I To prove your guess:I Exact computation in appropriate number field, orI Use genericity
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Γ is a triangle group!
The Poincare polyhedron theorem gives the followingpresentation:
〈G1,G2 |G 42 , (G1G2)3, (G2G1G2)3 〉
The group has index 2 in a (3, 3, 4)-triangle group.
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Deformations
Using real (3, 3, 4)-triangle groups, one gets a 1-parameterfamily of deformations, where the unipotent elementbecomes elliptic with eigenvalues (1, ζ, ζ), |ζ| = 1.
These deformations give complete spherical CR structures on(k, 1)-surgeries of the figure eight knot (with k ≥ 5).
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
FKR Census
Geometricstructures on theFigure Eight Knot
Complement
ICERM WorkshopExotic Geometric
Structures
Martin Deraux
The figure eightknot
Presentation
Holonomy
Prism picture
Tetrahedron picture
Spherical CRgeometry
Limit set
Complex hyperbolicgeometry
Central question
Known results
Main theorem
Horotube
Combinatorics
Triangle group
Surgery
Rank 1 boundaryunipotent
Other examples
Hyperbolic manifolds with low complexity (up to 3tetrahedra), and representations into PU(2, 1) with
Unipotent and Rank 1,
boundary holonomy.List from Falbel, Koseleff and Rouiller (2013):
m004 = 41 knot
m009
m015 = 52 knot
Same seems to work:
I Dirichlet domains with finitely many faces (forwell-chosen center)
I The image groups are triangle groups.