geometric structures on the figure eight knot complement [1cm] … · 2018-04-25 · icerm workshop...

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


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


The figure eight knot

Various pictures of 41:

K = figure eight knot


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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)


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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.


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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.


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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.


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Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


Prism picture

x2

x3 x3 x3

x3x3

x4

x4

x4x2

x1 x1

x1x1

x4

x4

x3 x2

x4

x2 x1


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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”.


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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).


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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)).


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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).


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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).


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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).


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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!)


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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?


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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).


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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).


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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.


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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.


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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.


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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).


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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


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Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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.


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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).


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


FKR Census


Complement


Structures

Martin Deraux


Presentation

Holonomy

Prism picture

Tetrahedron picture


Limit set


Central question

Known results

Main theorem

Horotube

Combinatorics

Triangle group

Surgery


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.

geometric structures on the figure eight knot complement [1cm] … · 2018-04-25 · icerm workshop...

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