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Flexibility and Rigidity of 3-DimensionalConvex Projective Structures
Sam Ballas
April 24, 2013Thesis Defense
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What is Convex Projective Geometry?
• Convex projective geometry is a generalization ofhyperbolic geometry.
• Retains many features of hyperbolic geometry.• No Mostow rigidity.
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What is Convex Projective Geometry?
• Convex projective geometry is a generalization ofhyperbolic geometry.
• Retains many features of hyperbolic geometry.
• No Mostow rigidity.
![Page 4: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/4.jpg)
What is Convex Projective Geometry?
• Convex projective geometry is a generalization ofhyperbolic geometry.
• Retains many features of hyperbolic geometry.• No Mostow rigidity.
![Page 5: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/5.jpg)
Projective Space
• There is a natural action of R× on Rn+1\0 by scaling.• Let RPn = P(Rn+1\0) be the quotient of this action.
• Alternatively, RPn is the space of lines in Rn+1
• A Projective Line is the projectivization of a 2-plane in Rn+1
• A Projective Hyperplane is the projectivization of ann−plane in Rn+1.
• The automorphism group of RPn isPGLn+1(R) := GLn+1(R)/R×.
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Projective Space
• There is a natural action of R× on Rn+1\0 by scaling.• Let RPn = P(Rn+1\0) be the quotient of this action.• Alternatively, RPn is the space of lines in Rn+1
• A Projective Line is the projectivization of a 2-plane in Rn+1
• A Projective Hyperplane is the projectivization of ann−plane in Rn+1.
• The automorphism group of RPn isPGLn+1(R) := GLn+1(R)/R×.
![Page 7: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/7.jpg)
Projective Space
• There is a natural action of R× on Rn+1\0 by scaling.• Let RPn = P(Rn+1\0) be the quotient of this action.• Alternatively, RPn is the space of lines in Rn+1
• A Projective Line is the projectivization of a 2-plane in Rn+1
• A Projective Hyperplane is the projectivization of ann−plane in Rn+1.
• The automorphism group of RPn isPGLn+1(R) := GLn+1(R)/R×.
![Page 8: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/8.jpg)
Projective Space
• There is a natural action of R× on Rn+1\0 by scaling.• Let RPn = P(Rn+1\0) be the quotient of this action.• Alternatively, RPn is the space of lines in Rn+1
• A Projective Line is the projectivization of a 2-plane in Rn+1
• A Projective Hyperplane is the projectivization of ann−plane in Rn+1.
• The automorphism group of RPn isPGLn+1(R) := GLn+1(R)/R×.
![Page 9: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/9.jpg)
Projective Space
• There is a natural action of R× on Rn+1\0 by scaling.• Let RPn = P(Rn+1\0) be the quotient of this action.• Alternatively, RPn is the space of lines in Rn+1
• A Projective Line is the projectivization of a 2-plane in Rn+1
• A Projective Hyperplane is the projectivization of ann−plane in Rn+1.
• The automorphism group of RPn isPGLn+1(R) := GLn+1(R)/R×.
![Page 10: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/10.jpg)
A Splitting of RPn
• Let H be a hyperplane in Rn+1.• H gives rise to a splitting of RPn = Rn t RPn−1 into an
affine part and an ideal part (inhomogeneous coordinates).
• RPn\P(H) is called an affine patch.
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A Splitting of RPn
• Let H be a hyperplane in Rn+1.• H gives rise to a splitting of RPn = Rn t RPn−1 into an
affine part and an ideal part (inhomogeneous coordinates).
• RPn\P(H) is called an affine patch.
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A Splitting of RPn
• Let H be a hyperplane in Rn+1.• H gives rise to a splitting of RPn = Rn t RPn−1 into an
affine part and an ideal part (inhomogeneous coordinates).
• RPn\P(H) is called an affine patch.
![Page 13: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/13.jpg)
The Klein Model
• Let 〈x , y〉 = x1y1 + . . .+ xnyn − xn+1yn+1be standard form of signature (n,1) on Rn+1.
• Let C = x ∈ Rn+1|〈x , x〉 < 0• P(C) is the Klein model of Hn.• In the affine patch defined by H it is a disk.
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Nice Properties of Hyperbolic Space
• Convex: Intersection with projective lines is connected.
• Properly Convex: Convex and closure is contained in anaffine patch⇐⇒ Disjoint from some projective hyperplane.
• Strictly Convex: Properly convex and boundary containsno non-trivial projective line segments.
Convex projective geometry focuses on the geometry ofproperly (sometimes stictly) convex domains.
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Nice Properties of Hyperbolic Space
• Convex: Intersection with projective lines is connected.• Properly Convex: Convex and closure is contained in an
affine patch⇐⇒ Disjoint from some projective hyperplane.
• Strictly Convex: Properly convex and boundary containsno non-trivial projective line segments.
Convex projective geometry focuses on the geometry ofproperly (sometimes stictly) convex domains.
![Page 16: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/16.jpg)
Nice Properties of Hyperbolic Space
• Convex: Intersection with projective lines is connected.• Properly Convex: Convex and closure is contained in an
affine patch⇐⇒ Disjoint from some projective hyperplane.• Strictly Convex: Properly convex and boundary contains
no non-trivial projective line segments.
Convex projective geometry focuses on the geometry ofproperly (sometimes stictly) convex domains.
![Page 17: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/17.jpg)
Nice Properties of Hyperbolic Space
• Convex: Intersection with projective lines is connected.• Properly Convex: Convex and closure is contained in an
affine patch⇐⇒ Disjoint from some projective hyperplane.• Strictly Convex: Properly convex and boundary contains
no non-trivial projective line segments.
Convex projective geometry focuses on the geometry ofproperly (sometimes stictly) convex domains.
![Page 18: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/18.jpg)
Hilbert MetricLet Ω be a properly convex set and PGL(Ω) be the projectiveautomorphisms preserving Ω.
Every properly convex set Ω admits a Hilbert metric given by
dΩ(x , y) = log[a, x ; y ,b] = log(|x − b| |y − a||x − a| |y − b|
)
• When Ω is an ellipsoid dΩ is twice the hyperbolic metric.• PGL(Ω) ≤ Isom(Ω) and equal when Ω is strictly convex.• Discrete subgroups of PGL(Ω) act properly discontinuously
on Ω.
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Hilbert MetricLet Ω be a properly convex set and PGL(Ω) be the projectiveautomorphisms preserving Ω.
Every properly convex set Ω admits a Hilbert metric given by
dΩ(x , y) = log[a, x ; y ,b] = log(|x − b| |y − a||x − a| |y − b|
)
• When Ω is an ellipsoid dΩ is twice the hyperbolic metric.• PGL(Ω) ≤ Isom(Ω) and equal when Ω is strictly convex.• Discrete subgroups of PGL(Ω) act properly discontinuously
on Ω.
![Page 20: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/20.jpg)
Hilbert MetricLet Ω be a properly convex set and PGL(Ω) be the projectiveautomorphisms preserving Ω.
Every properly convex set Ω admits a Hilbert metric given by
dΩ(x , y) = log[a, x ; y ,b] = log(|x − b| |y − a||x − a| |y − b|
)
• When Ω is an ellipsoid dΩ is twice the hyperbolic metric.
• PGL(Ω) ≤ Isom(Ω) and equal when Ω is strictly convex.• Discrete subgroups of PGL(Ω) act properly discontinuously
on Ω.
![Page 21: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/21.jpg)
Hilbert MetricLet Ω be a properly convex set and PGL(Ω) be the projectiveautomorphisms preserving Ω.
Every properly convex set Ω admits a Hilbert metric given by
dΩ(x , y) = log[a, x ; y ,b] = log(|x − b| |y − a||x − a| |y − b|
)
• When Ω is an ellipsoid dΩ is twice the hyperbolic metric.• PGL(Ω) ≤ Isom(Ω) and equal when Ω is strictly convex.
• Discrete subgroups of PGL(Ω) act properly discontinuouslyon Ω.
![Page 22: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/22.jpg)
Hilbert MetricLet Ω be a properly convex set and PGL(Ω) be the projectiveautomorphisms preserving Ω.
Every properly convex set Ω admits a Hilbert metric given by
dΩ(x , y) = log[a, x ; y ,b] = log(|x − b| |y − a||x − a| |y − b|
)
• When Ω is an ellipsoid dΩ is twice the hyperbolic metric.• PGL(Ω) ≤ Isom(Ω) and equal when Ω is strictly convex.• Discrete subgroups of PGL(Ω) act properly discontinuously
on Ω.
![Page 23: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/23.jpg)
Classification of Isometries
If Ω is an open properly convex then PGL(Ω) embeds inSL±
n+1(R) which allows us to talk about eigenvalues.
If γ ∈ PGL(Ω) then γ is1. elliptic if γ fixes a point in Ω,2. parabolic if γ acts freely on Ω and has all eigenvalues of
modulus 1, and3. hyperbolic otherwise
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Classification of Isometries
If Ω is an open properly convex then PGL(Ω) embeds inSL±
n+1(R) which allows us to talk about eigenvalues.If γ ∈ PGL(Ω) then γ is
1. elliptic if γ fixes a point in Ω,2. parabolic if γ acts freely on Ω and has all eigenvalues of
modulus 1, and3. hyperbolic otherwise
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Similarities to Hyperbolic Isometries
1. When Ω is an ellipsoid this classification is the same as thestandard classification of hyperbolic isometries.
2. When Ω is strictly convex parabolic isometries have aunique fixed point on ∂Ω.
3. When Ω is strictly convex, hyperbolic isometries have 2fixed points on ∂Ω and act by translation along the lineconnecting them.
4. When Ω is strictly convex, parabolic and hyperbolicelements in a common discrete subgroup do not sharefixed points.
5. When Ω is strictly convex, a discrete, torsion-free subgroupof elements fixing a geodesic is infinite cyclic.
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Similarities to Hyperbolic Isometries
1. When Ω is an ellipsoid this classification is the same as thestandard classification of hyperbolic isometries.
2. When Ω is strictly convex parabolic isometries have aunique fixed point on ∂Ω.
3. When Ω is strictly convex, hyperbolic isometries have 2fixed points on ∂Ω and act by translation along the lineconnecting them.
4. When Ω is strictly convex, parabolic and hyperbolicelements in a common discrete subgroup do not sharefixed points.
5. When Ω is strictly convex, a discrete, torsion-free subgroupof elements fixing a geodesic is infinite cyclic.
![Page 27: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/27.jpg)
Similarities to Hyperbolic Isometries
1. When Ω is an ellipsoid this classification is the same as thestandard classification of hyperbolic isometries.
2. When Ω is strictly convex parabolic isometries have aunique fixed point on ∂Ω.
3. When Ω is strictly convex, hyperbolic isometries have 2fixed points on ∂Ω and act by translation along the lineconnecting them.
4. When Ω is strictly convex, parabolic and hyperbolicelements in a common discrete subgroup do not sharefixed points.
5. When Ω is strictly convex, a discrete, torsion-free subgroupof elements fixing a geodesic is infinite cyclic.
![Page 28: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/28.jpg)
Similarities to Hyperbolic Isometries
1. When Ω is an ellipsoid this classification is the same as thestandard classification of hyperbolic isometries.
2. When Ω is strictly convex parabolic isometries have aunique fixed point on ∂Ω.
3. When Ω is strictly convex, hyperbolic isometries have 2fixed points on ∂Ω and act by translation along the lineconnecting them.
4. When Ω is strictly convex, parabolic and hyperbolicelements in a common discrete subgroup do not sharefixed points.
5. When Ω is strictly convex, a discrete, torsion-free subgroupof elements fixing a geodesic is infinite cyclic.
![Page 29: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/29.jpg)
Similarities to Hyperbolic Isometries
1. When Ω is an ellipsoid this classification is the same as thestandard classification of hyperbolic isometries.
2. When Ω is strictly convex parabolic isometries have aunique fixed point on ∂Ω.
3. When Ω is strictly convex, hyperbolic isometries have 2fixed points on ∂Ω and act by translation along the lineconnecting them.
4. When Ω is strictly convex, parabolic and hyperbolicelements in a common discrete subgroup do not sharefixed points.
5. When Ω is strictly convex, a discrete, torsion-free subgroupof elements fixing a geodesic is infinite cyclic.
![Page 30: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/30.jpg)
Convex Projective Manifolds
Let Mn be a manifold with π1(M) = Γ. A convex projectivestructure on M is a pair (Ω, ρ) such that
1. Ω is a properly convex open subset of RPn.2. ρ : Γ→ PGL(Ω) is a discrete and faithful representation.3. M ∼= Ω/ρ(Γ)
• ρ is called the holonomy of the structure• The structure is strictly convex if Ω is strictly convex• When Ω is an ellipsoid then PGL(Ω) ∼= Isom(Hn) and a
complete hyperbolic structure is a strictly convex projectivestructure.
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Convex Projective Manifolds
Let Mn be a manifold with π1(M) = Γ. A convex projectivestructure on M is a pair (Ω, ρ) such that
1. Ω is a properly convex open subset of RPn.2. ρ : Γ→ PGL(Ω) is a discrete and faithful representation.3. M ∼= Ω/ρ(Γ)
• ρ is called the holonomy of the structure
• The structure is strictly convex if Ω is strictly convex• When Ω is an ellipsoid then PGL(Ω) ∼= Isom(Hn) and a
complete hyperbolic structure is a strictly convex projectivestructure.
![Page 32: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/32.jpg)
Convex Projective Manifolds
Let Mn be a manifold with π1(M) = Γ. A convex projectivestructure on M is a pair (Ω, ρ) such that
1. Ω is a properly convex open subset of RPn.2. ρ : Γ→ PGL(Ω) is a discrete and faithful representation.3. M ∼= Ω/ρ(Γ)
• ρ is called the holonomy of the structure• The structure is strictly convex if Ω is strictly convex
• When Ω is an ellipsoid then PGL(Ω) ∼= Isom(Hn) and acomplete hyperbolic structure is a strictly convex projectivestructure.
![Page 33: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/33.jpg)
Convex Projective Manifolds
Let Mn be a manifold with π1(M) = Γ. A convex projectivestructure on M is a pair (Ω, ρ) such that
1. Ω is a properly convex open subset of RPn.2. ρ : Γ→ PGL(Ω) is a discrete and faithful representation.3. M ∼= Ω/ρ(Γ)
• ρ is called the holonomy of the structure• The structure is strictly convex if Ω is strictly convex• When Ω is an ellipsoid then PGL(Ω) ∼= Isom(Hn) and a
complete hyperbolic structure is a strictly convex projectivestructure.
![Page 34: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/34.jpg)
Projective EquivalenceSuppose that Mn ∼= Ωi/ρi(Γ) for i = 1,2, then (Ω1, ρ1) and(Ω2, ρ2) are projectively equivalent if there existsh ∈ PGLn+1(R) such that h(Ω1) = Ω2 and for each γ ∈ π1(M)
Ω1
ρ1(γ)
h // Ω2
ρ2(γ)
Ω1h // Ω2
• If (Ω1, ρ1) and (Ω2, ρ2) are projectively equivalent thenρ2(Γ) = hρ1(Γ)h−1
• Let X(Γ,PGLn+1(R)) be the set of conjugacy classes ofrepresentations from Γ to PGLn+1(R).
Projectiveequivalence classes of M are in bijective correspondencewith elements of X(Γ,PGLn+1(R)) that are faithful, discrete,and preserve a properly convex set.
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Projective EquivalenceSuppose that Mn ∼= Ωi/ρi(Γ) for i = 1,2, then (Ω1, ρ1) and(Ω2, ρ2) are projectively equivalent if there existsh ∈ PGLn+1(R) such that h(Ω1) = Ω2 and for each γ ∈ π1(M)
Ω1
ρ1(γ)
h // Ω2
ρ2(γ)
Ω1h // Ω2
• If (Ω1, ρ1) and (Ω2, ρ2) are projectively equivalent thenρ2(Γ) = hρ1(Γ)h−1
• Let X(Γ,PGLn+1(R)) be the set of conjugacy classes ofrepresentations from Γ to PGLn+1(R).
Projectiveequivalence classes of M are in bijective correspondencewith elements of X(Γ,PGLn+1(R)) that are faithful, discrete,and preserve a properly convex set.
![Page 36: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/36.jpg)
Projective EquivalenceSuppose that Mn ∼= Ωi/ρi(Γ) for i = 1,2, then (Ω1, ρ1) and(Ω2, ρ2) are projectively equivalent if there existsh ∈ PGLn+1(R) such that h(Ω1) = Ω2 and for each γ ∈ π1(M)
Ω1
ρ1(γ)
h // Ω2
ρ2(γ)
Ω1h // Ω2
• If (Ω1, ρ1) and (Ω2, ρ2) are projectively equivalent thenρ2(Γ) = hρ1(Γ)h−1
• Let X(Γ,PGLn+1(R)) be the set of conjugacy classes ofrepresentations from Γ to PGLn+1(R).
Projectiveequivalence classes of M are in bijective correspondencewith elements of X(Γ,PGLn+1(R)) that are faithful, discrete,and preserve a properly convex set.
![Page 37: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/37.jpg)
Projective EquivalenceSuppose that Mn ∼= Ωi/ρi(Γ) for i = 1,2, then (Ω1, ρ1) and(Ω2, ρ2) are projectively equivalent if there existsh ∈ PGLn+1(R) such that h(Ω1) = Ω2 and for each γ ∈ π1(M)
Ω1
ρ1(γ)
h // Ω2
ρ2(γ)
Ω1h // Ω2
• If (Ω1, ρ1) and (Ω2, ρ2) are projectively equivalent thenρ2(Γ) = hρ1(Γ)h−1
• Let X(Γ,PGLn+1(R)) be the set of conjugacy classes ofrepresentations from Γ to PGLn+1(R). Projectiveequivalence classes of M are in bijective correspondencewith elements of X(Γ,PGLn+1(R)) that are faithful, discrete,and preserve a properly convex set.
![Page 38: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/38.jpg)
Mostow Rigidity
Let Mn be a finite volume hyperbolic manifold (n ≥ 3) and let(Ω1, ρ1) and (Ω2, ρ2) be two complete hyperbolic structures onM. Mostow rigidity tells us that (Ω1, ρ1) and (Ω2, ρ2) areprojectively equivalent.
There is a distinguished projective equivalence class of convexprojective structures on M consisting of complete hyperbolicstructures on M.
![Page 39: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/39.jpg)
Mostow Rigidity
Let Mn be a finite volume hyperbolic manifold (n ≥ 3) and let(Ω1, ρ1) and (Ω2, ρ2) be two complete hyperbolic structures onM. Mostow rigidity tells us that (Ω1, ρ1) and (Ω2, ρ2) areprojectively equivalent.
There is a distinguished projective equivalence class of convexprojective structures on M consisting of complete hyperbolicstructures on M.
![Page 40: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/40.jpg)
Rigidity and FlexibilityQuestions
1. Are there other projective equivalence classes of (strictly)convex projective structures on M near the completehyperbolic structure?
Yes
• Dimension 2(Goldman-Choi)
• Bending (Johnson-Millson)
• Flexing (Cooper-Long-Thistlethwaite)
• Certain surgeries onFigure-8 (Huesener-Porti)
No
• Most closed 2-generatorcensus manifolds (Cooper-Long-Thistlethwaite)
2. How do we know when deformations exist?
![Page 41: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/41.jpg)
Rigidity and FlexibilityQuestions
1. Are there other projective equivalence classes of (strictly)convex projective structures on M near the completehyperbolic structure?
Yes
• Dimension 2(Goldman-Choi)
• Bending (Johnson-Millson)
• Flexing (Cooper-Long-Thistlethwaite)
• Certain surgeries onFigure-8 (Huesener-Porti)
No
• Most closed 2-generatorcensus manifolds (Cooper-Long-Thistlethwaite)
2. How do we know when deformations exist?
![Page 42: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/42.jpg)
Rigidity and FlexibilityQuestions
1. Are there other projective equivalence classes of (strictly)convex projective structures on M near the completehyperbolic structure?
Yes
• Dimension 2(Goldman-Choi)
• Bending (Johnson-Millson)
• Flexing (Cooper-Long-Thistlethwaite)
• Certain surgeries onFigure-8 (Huesener-Porti)
No
• Most closed 2-generatorcensus manifolds (Cooper-Long-Thistlethwaite)
2. How do we know when deformations exist?
![Page 43: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/43.jpg)
Rigidity and FlexibilityQuestions
1. Are there other projective equivalence classes of (strictly)convex projective structures on M near the completehyperbolic structure?
Yes
• Dimension 2(Goldman-Choi)
• Bending (Johnson-Millson)
• Flexing (Cooper-Long-Thistlethwaite)
• Certain surgeries onFigure-8 (Huesener-Porti)
No
• Most closed 2-generatorcensus manifolds (Cooper-Long-Thistlethwaite)
2. How do we know when deformations exist?
![Page 44: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/44.jpg)
Rigidity and FlexibilityQuestions
1. Are there other projective equivalence classes of (strictly)convex projective structures on M near the completehyperbolic structure?
Yes
• Dimension 2(Goldman-Choi)
• Bending (Johnson-Millson)
• Flexing (Cooper-Long-Thistlethwaite)
• Certain surgeries onFigure-8 (Huesener-Porti)
No
• Most closed 2-generatorcensus manifolds (Cooper-Long-Thistlethwaite)
2. How do we know when deformations exist?
![Page 45: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/45.jpg)
Rigidity and FlexibilityQuestions
1. Are there other projective equivalence classes of (strictly)convex projective structures on M near the completehyperbolic structure?
Yes
• Dimension 2(Goldman-Choi)
• Bending (Johnson-Millson)
• Flexing (Cooper-Long-Thistlethwaite)
• Certain surgeries onFigure-8 (Huesener-Porti)
No
• Most closed 2-generatorcensus manifolds (Cooper-Long-Thistlethwaite)
2. How do we know when deformations exist?
![Page 46: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/46.jpg)
Rigidity and FlexibilityQuestions
1. Are there other projective equivalence classes of (strictly)convex projective structures on M near the completehyperbolic structure?
Yes
• Dimension 2(Goldman-Choi)
• Bending (Johnson-Millson)
• Flexing (Cooper-Long-Thistlethwaite)
• Certain surgeries onFigure-8 (Huesener-Porti)
No
• Most closed 2-generatorcensus manifolds (Cooper-Long-Thistlethwaite)
2. How do we know when deformations exist?
![Page 47: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/47.jpg)
Rigidity and FlexibilityQuestions
1. Are there other projective equivalence classes of (strictly)convex projective structures on M near the completehyperbolic structure?
Yes
• Dimension 2(Goldman-Choi)
• Bending (Johnson-Millson)
• Flexing (Cooper-Long-Thistlethwaite)
• Certain surgeries onFigure-8 (Huesener-Porti)
No
• Most closed 2-generatorcensus manifolds (Cooper-Long-Thistlethwaite)
2. How do we know when deformations exist?
![Page 48: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/48.jpg)
A decomposition of M
Let M be an orientable, finite volume, hyperbolic 3-manifold.Then
M = MK ∪ (tiCi).
Ci∼= T 2 × [1,∞) are called cusps and π1(Ci) is a peripheral
subgroup.
• If ρ0 is the holonomy of the complete hyperbolic structureon M then T 2 × x has the same Euclidean structure foreach x ∈ [1,∞).
• If ρ1 is the holonomy of a general convex projectivestructure on M then T 2 × x has the same affine structurefor each x ∈ [1,∞).
![Page 49: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/49.jpg)
A decomposition of M
Let M be an orientable, finite volume, hyperbolic 3-manifold.Then
M = MK ∪ (tiCi).
Ci∼= T 2 × [1,∞) are called cusps and π1(Ci) is a peripheral
subgroup.• If ρ0 is the holonomy of the complete hyperbolic structure
on M then T 2 × x has the same Euclidean structure foreach x ∈ [1,∞).
• If ρ1 is the holonomy of a general convex projectivestructure on M then T 2 × x has the same affine structurefor each x ∈ [1,∞).
![Page 50: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/50.jpg)
A decomposition of M
Let M be an orientable, finite volume, hyperbolic 3-manifold.Then
M = MK ∪ (tiCi).
Ci∼= T 2 × [1,∞) are called cusps and π1(Ci) is a peripheral
subgroup.• If ρ0 is the holonomy of the complete hyperbolic structure
on M then T 2 × x has the same Euclidean structure foreach x ∈ [1,∞).
• If ρ1 is the holonomy of a general convex projectivestructure on M then T 2 × x has the same affine structurefor each x ∈ [1,∞).
![Page 51: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/51.jpg)
Description of the Holonomy
What does the holonomy of a strictly convex structure on Mlook like?
Lemma 1 (Cooper-Long-Tillman)Let Ω ⊂ RP3 be properly convex. If γ ∈ PGL(Ω) is parabolicthen γ is conjugate in PGL4(R) to
1 0 0 00 1 1 00 0 1 10 0 0 1
If γ ∈ PGL4(R) is conjugate the above matrix then we say that γis a strictly convex parabolic.
![Page 52: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/52.jpg)
Description of the Holonomy
What does the holonomy of a strictly convex structure on Mlook like?
Lemma 1 (Cooper-Long-Tillman)Let Ω ⊂ RP3 be properly convex. If γ ∈ PGL(Ω) is parabolicthen γ is conjugate in PGL4(R) to
1 0 0 00 1 1 00 0 1 10 0 0 1
If γ ∈ PGL4(R) is conjugate the above matrix then we say that γis a strictly convex parabolic.
![Page 53: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/53.jpg)
Description of the Holonomy
What does the holonomy of a strictly convex structure on Mlook like?
Lemma 1 (Cooper-Long-Tillman)Let Ω ⊂ RP3 be properly convex. If γ ∈ PGL(Ω) is parabolicthen γ is conjugate in PGL4(R) to
1 0 0 00 1 1 00 0 1 10 0 0 1
If γ ∈ PGL4(R) is conjugate the above matrix then we say that γis a strictly convex parabolic.
![Page 54: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/54.jpg)
Description of the Holonomy
Lemma 2If ρ is the holonomy of a strictly convex projective structure onM then ρ(π1(C)) is parabolic for each cusp C of M.
Let Xscp(Γ,PGL4(R)) be conjugacy classes of representationssuch that the image of every peripheral element is a strictlyconvex parabolic element.
Corollary 3If ρ is the holonomy of a strictly convex projective structure onM then [ρ] ∈ Xscp(Γ,PGL4(R))
![Page 55: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/55.jpg)
Description of the Holonomy
Lemma 2If ρ is the holonomy of a strictly convex projective structure onM then ρ(π1(C)) is parabolic for each cusp C of M.
Let Xscp(Γ,PGL4(R)) be conjugacy classes of representationssuch that the image of every peripheral element is a strictlyconvex parabolic element.
Corollary 3If ρ is the holonomy of a strictly convex projective structure onM then [ρ] ∈ Xscp(Γ,PGL4(R))
![Page 56: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/56.jpg)
Description of the Holonomy
Lemma 2If ρ is the holonomy of a strictly convex projective structure onM then ρ(π1(C)) is parabolic for each cusp C of M.
Let Xscp(Γ,PGL4(R)) be conjugacy classes of representationssuch that the image of every peripheral element is a strictlyconvex parabolic element.
Corollary 3If ρ is the holonomy of a strictly convex projective structure onM then [ρ] ∈ Xscp(Γ,PGL4(R))
![Page 57: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/57.jpg)
Two-Bridge Knots
If M is a two bridge knot complement thenΓ = π1(M) = 〈α, β|αω = ωβ〉, where ω is a word in α and β thatdepends on the knot.
• α and β can be taken to be meridians• We want to look for ρ : Γ→ PGL4(R) where α and β are
sent to strictly convex parabolic elements
![Page 58: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/58.jpg)
Two-Bridge Knots
If M is a two bridge knot complement thenΓ = π1(M) = 〈α, β|αω = ωβ〉, where ω is a word in α and β thatdepends on the knot.
• α and β can be taken to be meridians
• We want to look for ρ : Γ→ PGL4(R) where α and β aresent to strictly convex parabolic elements
![Page 59: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/59.jpg)
Two-Bridge Knots
If M is a two bridge knot complement thenΓ = π1(M) = 〈α, β|αω = ωβ〉, where ω is a word in α and β thatdepends on the knot.
• α and β can be taken to be meridians• We want to look for ρ : Γ→ PGL4(R) where α and β are
sent to strictly convex parabolic elements
![Page 60: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/60.jpg)
A Normal Form
By work of Riley it is possible to uniquely conjugatenon-commuting parabolic a,b ∈ Isom(H3) ∼= PSL2(C) so that
a =
(1 10 1
), b =
(1 0z 1
),
where z is a non-zero complex number.
Geometrically, this is done be moving the repsective fixedpoints of a and b to∞ and 0
![Page 61: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/61.jpg)
A Normal Form
By work of Riley it is possible to uniquely conjugatenon-commuting parabolic a,b ∈ Isom(H3) ∼= PSL2(C) so that
a =
(1 10 1
), b =
(1 0z 1
),
where z is a non-zero complex number.Geometrically, this is done be moving the repsective fixedpoints of a and b to∞ and 0
![Page 62: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/62.jpg)
A Normal Form
Let ρ be a strictly convex holonomy near the completehyperbolic holonomy. Let Eα and Eβ be the 1-eigenspaces ofρ(α) and ρ(β).
By irreduciblity, R4 = Eα ⊕ Eβ and so we can find a basis where
ρ(α) =
(I Au0 Al
), ρ(β) =
(Bu 0Bl I
)
The minimal polynomial of a strictly convex parabolic is(x − 1)3. Therefore, neither Al and Bu are diagonalizable andso by further conjugating we can assume that
Al =
(1 a30 1
), Bu =
(1 0b1 1
)
![Page 63: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/63.jpg)
A Normal Form
Let ρ be a strictly convex holonomy near the completehyperbolic holonomy. Let Eα and Eβ be the 1-eigenspaces ofρ(α) and ρ(β).By irreduciblity, R4 = Eα ⊕ Eβ and so we can find a basis where
ρ(α) =
(I Au0 Al
), ρ(β) =
(Bu 0Bl I
)
The minimal polynomial of a strictly convex parabolic is(x − 1)3. Therefore, neither Al and Bu are diagonalizable andso by further conjugating we can assume that
Al =
(1 a30 1
), Bu =
(1 0b1 1
)
![Page 64: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/64.jpg)
A Normal Form
Let ρ be a strictly convex holonomy near the completehyperbolic holonomy. Let Eα and Eβ be the 1-eigenspaces ofρ(α) and ρ(β).By irreduciblity, R4 = Eα ⊕ Eβ and so we can find a basis where
ρ(α) =
(I Au0 Al
), ρ(β) =
(Bu 0Bl I
)
The minimal polynomial of a strictly convex parabolic is(x − 1)3. Therefore, neither Al and Bu are diagonalizable andso by further conjugating we can assume that
Al =
(1 a30 1
), Bu =
(1 0b1 1
)
![Page 65: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/65.jpg)
A Normal FormConjugacies that preserve this form look like
u11 0 0 0u21 u22 0 00 0 u33 u340 0 0 u44
Therefore we can uniquely conjugate so that
ρ(α) =
1 0 1 a10 1 1 a20 0 1 a30 0 0 1
, ρ(β) =
1 0 0 0b1 1 0 0b2 1 1 01 1 0 1
Each solution to the matrix equation ρ(α)ρ(ω)− ρ(ω)ρ(β) = 0gives a conjugacy class of representations for the two bridgeknot complement.
![Page 66: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/66.jpg)
A Normal FormConjugacies that preserve this form look like
u11 0 0 0u21 u22 0 00 0 u33 u340 0 0 u44
Therefore we can uniquely conjugate so that
ρ(α) =
1 0 1 a10 1 1 a20 0 1 a30 0 0 1
, ρ(β) =
1 0 0 0b1 1 0 0b2 1 1 01 1 0 1
Each solution to the matrix equation ρ(α)ρ(ω)− ρ(ω)ρ(β) = 0gives a conjugacy class of representations for the two bridgeknot complement.
![Page 67: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/67.jpg)
A Normal FormConjugacies that preserve this form look like
u11 0 0 0u21 u22 0 00 0 u33 u340 0 0 u44
Therefore we can uniquely conjugate so that
ρ(α) =
1 0 1 a10 1 1 a20 0 1 a30 0 0 1
, ρ(β) =
1 0 0 0b1 1 0 0b2 1 1 01 1 0 1
Each solution to the matrix equation ρ(α)ρ(ω)− ρ(ω)ρ(β) = 0gives a conjugacy class of representations for the two bridgeknot complement.
![Page 68: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/68.jpg)
Figure-8 Example
Let M be the figure-8 knot complement, then ω = βα−1β−1αand solutions to the previous equation are
ρt (α) =
1 0 1 3−t
t−20 1 1 1
2(t−2)
0 0 1 t2(t−2)
0 0 0 1
, ρt (β) =
1 0 0 0t 1 0 02 1 1 01 1 0 1
,
and the complete hyperbolic structure occurs at t = 4.• The element l = βα−1β−1α2β−1α−1β is a longitude andρt (l) is parabolic iff t = 4.
• Locally, the complete hyperbolic structure is the uniquestrictly convex projective structure on M
![Page 69: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/69.jpg)
Figure-8 Example
Let M be the figure-8 knot complement, then ω = βα−1β−1αand solutions to the previous equation are
ρt (α) =
1 0 1 3−t
t−20 1 1 1
2(t−2)
0 0 1 t2(t−2)
0 0 0 1
, ρt (β) =
1 0 0 0t 1 0 02 1 1 01 1 0 1
,
and the complete hyperbolic structure occurs at t = 4.
• The element l = βα−1β−1α2β−1α−1β is a longitude andρt (l) is parabolic iff t = 4.
• Locally, the complete hyperbolic structure is the uniquestrictly convex projective structure on M
![Page 70: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/70.jpg)
Figure-8 Example
Let M be the figure-8 knot complement, then ω = βα−1β−1αand solutions to the previous equation are
ρt (α) =
1 0 1 3−t
t−20 1 1 1
2(t−2)
0 0 1 t2(t−2)
0 0 0 1
, ρt (β) =
1 0 0 0t 1 0 02 1 1 01 1 0 1
,
and the complete hyperbolic structure occurs at t = 4.• The element l = βα−1β−1α2β−1α−1β is a longitude andρt (l) is parabolic iff t = 4.
• Locally, the complete hyperbolic structure is the uniquestrictly convex projective structure on M
![Page 71: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/71.jpg)
Figure-8 Example
Let M be the figure-8 knot complement, then ω = βα−1β−1αand solutions to the previous equation are
ρt (α) =
1 0 1 3−t
t−20 1 1 1
2(t−2)
0 0 1 t2(t−2)
0 0 0 1
, ρt (β) =
1 0 0 0t 1 0 02 1 1 01 1 0 1
,
and the complete hyperbolic structure occurs at t = 4.• The element l = βα−1β−1α2β−1α−1β is a longitude andρt (l) is parabolic iff t = 4.
• Locally, the complete hyperbolic structure is the uniquestrictly convex projective structure on M
![Page 72: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/72.jpg)
Other Two-bridge Knots and Links
• There are similar rigidity results for the knots 52, 61, andthe Whitehead link.
• In these other cases there are no families ofrepresentations where ρ(α) and ρ(β) are parabolic.
(this is likely because of amphicheirality of the figure-8)
• There is strong numerical evidence that several othertwo-bridge knots are rigid.
• Is there a general rigidity result for two-bridge knots andlinks?
![Page 73: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/73.jpg)
Other Two-bridge Knots and Links
• There are similar rigidity results for the knots 52, 61, andthe Whitehead link.
• In these other cases there are no families ofrepresentations where ρ(α) and ρ(β) are parabolic.(this is likely because of amphicheirality of the figure-8)
• There is strong numerical evidence that several othertwo-bridge knots are rigid.
• Is there a general rigidity result for two-bridge knots andlinks?
![Page 74: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/74.jpg)
Other Two-bridge Knots and Links
• There are similar rigidity results for the knots 52, 61, andthe Whitehead link.
• In these other cases there are no families ofrepresentations where ρ(α) and ρ(β) are parabolic.(this is likely because of amphicheirality of the figure-8)
• There is strong numerical evidence that several othertwo-bridge knots are rigid.
• Is there a general rigidity result for two-bridge knots andlinks?
![Page 75: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/75.jpg)
Other Two-bridge Knots and Links
• There are similar rigidity results for the knots 52, 61, andthe Whitehead link.
• In these other cases there are no families ofrepresentations where ρ(α) and ρ(β) are parabolic.(this is likely because of amphicheirality of the figure-8)
• There is strong numerical evidence that several othertwo-bridge knots are rigid.
• Is there a general rigidity result for two-bridge knots andlinks?
![Page 76: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/76.jpg)
Finding DeformationsThe Closed Case
Let M be a closed manifold (or orbifold). Which deformations ofrepresentations give rise to strictly convex projectivestructures?
Theorem 4 (Koszul, Benoist)Let M be a closed, hyperbolic 3-manifold and ρ0 be theholonomy of the complete hyperbolic structure on M. If ρt issufficiently close to ρ0 in Hom(Γ,PGL4(R)) then ρt is theholonomy of a strictly convex projective structure on M
• Small deformations of holonomy correspond to smalldeformations of the convex projective structure
• To find deformations of convex projective structures weonly need to deform the conjugacy class ofrepresentations.
![Page 77: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/77.jpg)
Finding DeformationsThe Closed Case
Let M be a closed manifold (or orbifold). Which deformations ofrepresentations give rise to strictly convex projectivestructures?
Theorem 4 (Koszul, Benoist)Let M be a closed, hyperbolic 3-manifold and ρ0 be theholonomy of the complete hyperbolic structure on M. If ρt issufficiently close to ρ0 in Hom(Γ,PGL4(R)) then ρt is theholonomy of a strictly convex projective structure on M
• Small deformations of holonomy correspond to smalldeformations of the convex projective structure
• To find deformations of convex projective structures weonly need to deform the conjugacy class ofrepresentations.
![Page 78: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/78.jpg)
Finding DeformationsThe Closed Case
Let M be a closed manifold (or orbifold). Which deformations ofrepresentations give rise to strictly convex projectivestructures?
Theorem 4 (Koszul, Benoist)Let M be a closed, hyperbolic 3-manifold and ρ0 be theholonomy of the complete hyperbolic structure on M. If ρt issufficiently close to ρ0 in Hom(Γ,PGL4(R)) then ρt is theholonomy of a strictly convex projective structure on M
• Small deformations of holonomy correspond to smalldeformations of the convex projective structure
• To find deformations of convex projective structures weonly need to deform the conjugacy class ofrepresentations.
![Page 79: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/79.jpg)
Finding DeformationsThe Closed Case
Let M be a closed manifold (or orbifold). Which deformations ofrepresentations give rise to strictly convex projectivestructures?
Theorem 4 (Koszul, Benoist)Let M be a closed, hyperbolic 3-manifold and ρ0 be theholonomy of the complete hyperbolic structure on M. If ρt issufficiently close to ρ0 in Hom(Γ,PGL4(R)) then ρt is theholonomy of a strictly convex projective structure on M
• Small deformations of holonomy correspond to smalldeformations of the convex projective structure
• To find deformations of convex projective structures weonly need to deform the conjugacy class ofrepresentations.
![Page 80: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/80.jpg)
Group Cohomology
Let ρt : Γ→ PGL4(R) be a representation, then for γ ∈ Γ andt ∈ (−ε, ε) we have
ρt (γ) = (I + z1(γ)t + z2(γ)t2 + . . .)ρ0(γ),
where zi : Γ→ sl4 are 1-cochain.
• The homomorphism condition tells us that z1 is a 1-cocylein twisted group cohomology.
• If ρt (γ) = ctρ0(γ)c−1t , then z1 is a 1-coboundary.
• H1(Γ) infinitesimally parametrizes conjugacy classes ofdeformations.
• Dimension of H1(Γ) gives an upper bound on thedimension of X(Γ,PGL4(R))
![Page 81: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/81.jpg)
Group Cohomology
Let ρt : Γ→ PGL4(R) be a representation, then for γ ∈ Γ andt ∈ (−ε, ε) we have
ρt (γ) = (I + z1(γ)t + z2(γ)t2 + . . .)ρ0(γ),
where zi : Γ→ sl4 are 1-cochain.• The homomorphism condition tells us that z1 is a 1-cocyle
in twisted group cohomology.
• If ρt (γ) = ctρ0(γ)c−1t , then z1 is a 1-coboundary.
• H1(Γ) infinitesimally parametrizes conjugacy classes ofdeformations.
• Dimension of H1(Γ) gives an upper bound on thedimension of X(Γ,PGL4(R))
![Page 82: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/82.jpg)
Group Cohomology
Let ρt : Γ→ PGL4(R) be a representation, then for γ ∈ Γ andt ∈ (−ε, ε) we have
ρt (γ) = (I + z1(γ)t + z2(γ)t2 + . . .)ρ0(γ),
where zi : Γ→ sl4 are 1-cochain.• The homomorphism condition tells us that z1 is a 1-cocyle
in twisted group cohomology.• If ρt (γ) = ctρ0(γ)c−1
t , then z1 is a 1-coboundary.
• H1(Γ) infinitesimally parametrizes conjugacy classes ofdeformations.
• Dimension of H1(Γ) gives an upper bound on thedimension of X(Γ,PGL4(R))
![Page 83: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/83.jpg)
Group Cohomology
Let ρt : Γ→ PGL4(R) be a representation, then for γ ∈ Γ andt ∈ (−ε, ε) we have
ρt (γ) = (I + z1(γ)t + z2(γ)t2 + . . .)ρ0(γ),
where zi : Γ→ sl4 are 1-cochain.• The homomorphism condition tells us that z1 is a 1-cocyle
in twisted group cohomology.• If ρt (γ) = ctρ0(γ)c−1
t , then z1 is a 1-coboundary.• H1(Γ) infinitesimally parametrizes conjugacy classes of
deformations.
• Dimension of H1(Γ) gives an upper bound on thedimension of X(Γ,PGL4(R))
![Page 84: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/84.jpg)
Group Cohomology
Let ρt : Γ→ PGL4(R) be a representation, then for γ ∈ Γ andt ∈ (−ε, ε) we have
ρt (γ) = (I + z1(γ)t + z2(γ)t2 + . . .)ρ0(γ),
where zi : Γ→ sl4 are 1-cochain.• The homomorphism condition tells us that z1 is a 1-cocyle
in twisted group cohomology.• If ρt (γ) = ctρ0(γ)c−1
t , then z1 is a 1-coboundary.• H1(Γ) infinitesimally parametrizes conjugacy classes of
deformations.• Dimension of H1(Γ) gives an upper bound on the
dimension of X(Γ,PGL4(R))
![Page 85: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/85.jpg)
Building Representations
Let ρt : Γ→ PGL4(R) be a representation, then for γ ∈ Γ andt ∈ (−ε, ε) we have
ρt (γ) = (I + z1(γ)t + z2(γ)t2 + . . .)ρ0(γ)
• The homomorphism condition also says that
k−1∑i=1
zi ∪ zk−i = dzk
• By a result of Artin, if we can find zi satisfying the abovecondition then we can build a convergent family ofrepresentations.
![Page 86: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/86.jpg)
Building Representations
Let ρt : Γ→ PGL4(R) be a representation, then for γ ∈ Γ andt ∈ (−ε, ε) we have
ρt (γ) = (I + z1(γ)t + z2(γ)t2 + . . .)ρ0(γ)
• The homomorphism condition also says that
k−1∑i=1
zi ∪ zk−i = dzk
• By a result of Artin, if we can find zi satisfying the abovecondition then we can build a convergent family ofrepresentations.
![Page 87: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/87.jpg)
Building Representations
Let ρt : Γ→ PGL4(R) be a representation, then for γ ∈ Γ andt ∈ (−ε, ε) we have
ρt (γ) = (I + z1(γ)t + z2(γ)t2 + . . .)ρ0(γ)
• The homomorphism condition also says that
k−1∑i=1
zi ∪ zk−i = dzk
• By a result of Artin, if we can find zi satisfying the abovecondition then we can build a convergent family ofrepresentations.
![Page 88: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/88.jpg)
Orbifold Surgery
Let M be the complement of an amphicheiral, hyperbolic knot,On be the orbifold obtained by the above gluing, andΓn = πorb
1 (On).
• By amphicheirality, there is a map φ : M → M s.tφ(m) = m−1 and φ(l) = l .
• φ extends to a symmetry φ : On → On
• We can use this symmetry to build representationsρt : Γn → PGL4(R)
![Page 89: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/89.jpg)
Orbifold Surgery
Let M be the complement of an amphicheiral, hyperbolic knot,On be the orbifold obtained by the above gluing, andΓn = πorb
1 (On).• By amphicheirality, there is a map φ : M → M s.tφ(m) = m−1 and φ(l) = l .
• φ extends to a symmetry φ : On → On
• We can use this symmetry to build representationsρt : Γn → PGL4(R)
![Page 90: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/90.jpg)
Orbifold Surgery
Let M be the complement of an amphicheiral, hyperbolic knot,On be the orbifold obtained by the above gluing, andΓn = πorb
1 (On).• By amphicheirality, there is a map φ : M → M s.tφ(m) = m−1 and φ(l) = l .
• φ extends to a symmetry φ : On → On
• We can use this symmetry to build representationsρt : Γn → PGL4(R)
![Page 91: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/91.jpg)
Orbifold Surgery
Let M be the complement of an amphicheiral, hyperbolic knot,On be the orbifold obtained by the above gluing, andΓn = πorb
1 (On).• By amphicheirality, there is a map φ : M → M s.tφ(m) = m−1 and φ(l) = l .
• φ extends to a symmetry φ : On → On
• We can use this symmetry to build representationsρt : Γn → PGL4(R)
![Page 92: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/92.jpg)
A Flexibility Theorem
Theorem 5 (B)Let M be the complement of a hyperbolic, amphicheiral knot,and suppose that M is infinitesimally projectively rigid relative tothe boundary at the complete hyperbolic structure and thelongitude is a rigid slope. Then for sufficiently large n, On has aone dimensional space of strictly convex projectivedeformations near the complete hyperbolic structure.
![Page 93: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/93.jpg)
Finding the CochainsLet H1(On) and H2(On) be the first two cellular cohomologygroups with twisted coefficients for On.
Claim: H1(On) and H2(On) are 1-dimensional and φ∗ acts onthem by ±1 respectively.By Mayer-Vietoris we have
0→ H1(On)ι∗1 ⊕ι∗2→
1
H1(M) ⊕1
H1(N)ι∗3 −ι∗4→
2
H1(∂M)∼=1
E1 ⊕1
E−1∆∗→ H2(On)→ 0
Can show that
H1(On)ι∗3ι
∗1∼= E1
and
E−1∆∗∼= H2(On)
![Page 94: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/94.jpg)
Finding the CochainsLet H1(On) and H2(On) be the first two cellular cohomologygroups with twisted coefficients for On.Claim: H1(On) and H2(On) are 1-dimensional and φ∗ acts onthem by ±1 respectively.
By Mayer-Vietoris we have
0→ H1(On)ι∗1 ⊕ι∗2→
1
H1(M) ⊕1
H1(N)ι∗3 −ι∗4→
2
H1(∂M)∼=1
E1 ⊕1
E−1∆∗→ H2(On)→ 0
Can show that
H1(On)ι∗3ι
∗1∼= E1
and
E−1∆∗∼= H2(On)
![Page 95: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/95.jpg)
Finding the CochainsLet H1(On) and H2(On) be the first two cellular cohomologygroups with twisted coefficients for On.Claim: H1(On) and H2(On) are 1-dimensional and φ∗ acts onthem by ±1 respectively.By Mayer-Vietoris we have
0→ H1(On)ι∗1 ⊕ι∗2→
1
H1(M) ⊕1
H1(N)ι∗3 −ι∗4→
2
H1(∂M)∼=1
E1 ⊕1
E−1∆∗→ H2(On)→ 0
Can show that
H1(On)ι∗3ι
∗1∼= E1
and
E−1∆∗∼= H2(On)
![Page 96: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/96.jpg)
Finding the CochainsLet H1(On) and H2(On) be the first two cellular cohomologygroups with twisted coefficients for On.Claim: H1(On) and H2(On) are 1-dimensional and φ∗ acts onthem by ±1 respectively.By Mayer-Vietoris we have
0→ H1(On)ι∗1 ⊕ι∗2→
1
H1(M) ⊕1
H1(N)ι∗3 −ι∗4→
2
H1(∂M)∼=1
E1 ⊕1
E−1∆∗→ H2(On)→ 0
Can show that
H1(On)ι∗3ι
∗1∼= E1
and
E−1∆∗∼= H2(On)
![Page 97: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/97.jpg)
Finding the CochainsLet H1(On) and H2(On) be the first two cellular cohomologygroups with twisted coefficients for On.Claim: H1(On) and H2(On) are 1-dimensional and φ∗ acts onthem by ±1 respectively.By Mayer-Vietoris we have
0→ H1(On)ι∗1 ⊕ι∗2→
1
H1(M) ⊕1
H1(N)ι∗3 −ι∗4→
2
H1(∂M)∼=1
E1 ⊕1
E−1∆∗→ H2(On)→ 0
Can show that
H1(On)ι∗3ι
∗1∼= E1
and
E−1∆∗∼= H2(On)
![Page 98: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/98.jpg)
Finding the Cochains
Let [z1] ∈ H1(On) be a generator and assume that φ has orderK .
• Replace z1 with z∗1 = 1
K (z1 + φ∗(z1) + . . . (φ∗)K−1(z1))
• z1 ∪ z1 = φ∗(z1) ∪ φ∗(z1) = φ∗(z1 ∪ z1) ∼ −z1 ∪ z1
• [z1 ∪ z1] = 0 and there is z2 s.t. dz2 = z1 ∪ z1.• Replace z2 with z∗
2 .• z1 ∪ z2 + z2 ∪ z1 = φ∗(z1) ∪ φ∗(z2) + φ∗(z2) ∪ φ∗(z1) =φ∗(z1 ∪ z2 + z2 ∪ z1) ∼ −(z1 ∪ z2 + z2 ∪ z1)
• [z1 ∪ z2 + z2 ∪ z1] = 0 and there is z3 s.t.dz3 = z1 ∪ z2 + z2 ∪ z1
• Repeat indefinitely to get remaining zi .
![Page 99: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/99.jpg)
Finding the Cochains
Let [z1] ∈ H1(On) be a generator and assume that φ has orderK .• Replace z1 with z∗
1 = 1K (z1 + φ∗(z1) + . . . (φ∗)K−1(z1))
• z1 ∪ z1 = φ∗(z1) ∪ φ∗(z1) = φ∗(z1 ∪ z1) ∼ −z1 ∪ z1
• [z1 ∪ z1] = 0 and there is z2 s.t. dz2 = z1 ∪ z1.• Replace z2 with z∗
2 .• z1 ∪ z2 + z2 ∪ z1 = φ∗(z1) ∪ φ∗(z2) + φ∗(z2) ∪ φ∗(z1) =φ∗(z1 ∪ z2 + z2 ∪ z1) ∼ −(z1 ∪ z2 + z2 ∪ z1)
• [z1 ∪ z2 + z2 ∪ z1] = 0 and there is z3 s.t.dz3 = z1 ∪ z2 + z2 ∪ z1
• Repeat indefinitely to get remaining zi .
![Page 100: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/100.jpg)
Finding the Cochains
Let [z1] ∈ H1(On) be a generator and assume that φ has orderK .• Replace z1 with z∗
1 = 1K (z1 + φ∗(z1) + . . . (φ∗)K−1(z1))
• z1 ∪ z1 = φ∗(z1) ∪ φ∗(z1) = φ∗(z1 ∪ z1) ∼ −z1 ∪ z1
• [z1 ∪ z1] = 0 and there is z2 s.t. dz2 = z1 ∪ z1.• Replace z2 with z∗
2 .• z1 ∪ z2 + z2 ∪ z1 = φ∗(z1) ∪ φ∗(z2) + φ∗(z2) ∪ φ∗(z1) =φ∗(z1 ∪ z2 + z2 ∪ z1) ∼ −(z1 ∪ z2 + z2 ∪ z1)
• [z1 ∪ z2 + z2 ∪ z1] = 0 and there is z3 s.t.dz3 = z1 ∪ z2 + z2 ∪ z1
• Repeat indefinitely to get remaining zi .
![Page 101: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/101.jpg)
Finding the Cochains
Let [z1] ∈ H1(On) be a generator and assume that φ has orderK .• Replace z1 with z∗
1 = 1K (z1 + φ∗(z1) + . . . (φ∗)K−1(z1))
• z1 ∪ z1 = φ∗(z1) ∪ φ∗(z1) = φ∗(z1 ∪ z1) ∼ −z1 ∪ z1
• [z1 ∪ z1] = 0 and there is z2 s.t. dz2 = z1 ∪ z1.
• Replace z2 with z∗2 .
• z1 ∪ z2 + z2 ∪ z1 = φ∗(z1) ∪ φ∗(z2) + φ∗(z2) ∪ φ∗(z1) =φ∗(z1 ∪ z2 + z2 ∪ z1) ∼ −(z1 ∪ z2 + z2 ∪ z1)
• [z1 ∪ z2 + z2 ∪ z1] = 0 and there is z3 s.t.dz3 = z1 ∪ z2 + z2 ∪ z1
• Repeat indefinitely to get remaining zi .
![Page 102: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/102.jpg)
Finding the Cochains
Let [z1] ∈ H1(On) be a generator and assume that φ has orderK .• Replace z1 with z∗
1 = 1K (z1 + φ∗(z1) + . . . (φ∗)K−1(z1))
• z1 ∪ z1 = φ∗(z1) ∪ φ∗(z1) = φ∗(z1 ∪ z1) ∼ −z1 ∪ z1
• [z1 ∪ z1] = 0 and there is z2 s.t. dz2 = z1 ∪ z1.• Replace z2 with z∗
2 .
• z1 ∪ z2 + z2 ∪ z1 = φ∗(z1) ∪ φ∗(z2) + φ∗(z2) ∪ φ∗(z1) =φ∗(z1 ∪ z2 + z2 ∪ z1) ∼ −(z1 ∪ z2 + z2 ∪ z1)
• [z1 ∪ z2 + z2 ∪ z1] = 0 and there is z3 s.t.dz3 = z1 ∪ z2 + z2 ∪ z1
• Repeat indefinitely to get remaining zi .
![Page 103: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/103.jpg)
Finding the Cochains
Let [z1] ∈ H1(On) be a generator and assume that φ has orderK .• Replace z1 with z∗
1 = 1K (z1 + φ∗(z1) + . . . (φ∗)K−1(z1))
• z1 ∪ z1 = φ∗(z1) ∪ φ∗(z1) = φ∗(z1 ∪ z1) ∼ −z1 ∪ z1
• [z1 ∪ z1] = 0 and there is z2 s.t. dz2 = z1 ∪ z1.• Replace z2 with z∗
2 .• z1 ∪ z2 + z2 ∪ z1 = φ∗(z1) ∪ φ∗(z2) + φ∗(z2) ∪ φ∗(z1) =φ∗(z1 ∪ z2 + z2 ∪ z1) ∼ −(z1 ∪ z2 + z2 ∪ z1)
• [z1 ∪ z2 + z2 ∪ z1] = 0 and there is z3 s.t.dz3 = z1 ∪ z2 + z2 ∪ z1
• Repeat indefinitely to get remaining zi .
![Page 104: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/104.jpg)
Finding the Cochains
Let [z1] ∈ H1(On) be a generator and assume that φ has orderK .• Replace z1 with z∗
1 = 1K (z1 + φ∗(z1) + . . . (φ∗)K−1(z1))
• z1 ∪ z1 = φ∗(z1) ∪ φ∗(z1) = φ∗(z1 ∪ z1) ∼ −z1 ∪ z1
• [z1 ∪ z1] = 0 and there is z2 s.t. dz2 = z1 ∪ z1.• Replace z2 with z∗
2 .• z1 ∪ z2 + z2 ∪ z1 = φ∗(z1) ∪ φ∗(z2) + φ∗(z2) ∪ φ∗(z1) =φ∗(z1 ∪ z2 + z2 ∪ z1) ∼ −(z1 ∪ z2 + z2 ∪ z1)
• [z1 ∪ z2 + z2 ∪ z1] = 0 and there is z3 s.t.dz3 = z1 ∪ z2 + z2 ∪ z1
• Repeat indefinitely to get remaining zi .
![Page 105: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/105.jpg)
Finding the Cochains
Let [z1] ∈ H1(On) be a generator and assume that φ has orderK .• Replace z1 with z∗
1 = 1K (z1 + φ∗(z1) + . . . (φ∗)K−1(z1))
• z1 ∪ z1 = φ∗(z1) ∪ φ∗(z1) = φ∗(z1 ∪ z1) ∼ −z1 ∪ z1
• [z1 ∪ z1] = 0 and there is z2 s.t. dz2 = z1 ∪ z1.• Replace z2 with z∗
2 .• z1 ∪ z2 + z2 ∪ z1 = φ∗(z1) ∪ φ∗(z2) + φ∗(z2) ∪ φ∗(z1) =φ∗(z1 ∪ z2 + z2 ∪ z1) ∼ −(z1 ∪ z2 + z2 ∪ z1)
• [z1 ∪ z2 + z2 ∪ z1] = 0 and there is z3 s.t.dz3 = z1 ∪ z2 + z2 ∪ z1
• Repeat indefinitely to get remaining zi .
![Page 106: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/106.jpg)
Consequences
• There are many flexible examples given by takingbranched covers of the figure-8 knot
• There is strong numerical evidence that 63 satisfies thehypotheses of the theorem and gives rise to moreexamples.
• There are infinitely many amphicheiral two-bridge knots.
![Page 107: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/107.jpg)
Consequences
• There are many flexible examples given by takingbranched covers of the figure-8 knot
• There is strong numerical evidence that 63 satisfies thehypotheses of the theorem and gives rise to moreexamples.
• There are infinitely many amphicheiral two-bridge knots.
![Page 108: Flexibility and Rigidity of 3-Dimensional Convex ...web.math.ucsb.edu/~sballas/research/documents/Thesis_Defense.pdf · Convex projective geometry focuses on the geometry of properly](https://reader034.vdocuments.us/reader034/viewer/2022050313/5f75a7c470f75f3bcc3a76a5/html5/thumbnails/108.jpg)
Consequences
• There are many flexible examples given by takingbranched covers of the figure-8 knot
• There is strong numerical evidence that 63 satisfies thehypotheses of the theorem and gives rise to moreexamples.
• There are infinitely many amphicheiral two-bridge knots.