The Structure of Properly Convex...
Transcript of The Structure of Properly Convex...
![Page 1: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/1.jpg)
The Structure of Properly Convex Manifolds
Sam Ballas(joint with D. Long)
Workshop on Geometric Structures and Discrete GroupsAustin, TX
May 3, 2014
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Some Questions
• What are convex projective manifolds?
Generalizations of Hyperbolic manifolds• How are they similar to hyperbolic manifolds? How are
they different?
Strictly Convex⇒ very similar. Properly convex⇒ lesssimilar
• What sort of structure do convex projective manifoldshave?
Deformations of finite volume strictly convex manifolds arestructurally similar to complete finite volume hyperbolicmanifolds
![Page 3: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/3.jpg)
Some Questions
• What are convex projective manifolds?
Generalizations of Hyperbolic manifolds
• How are they similar to hyperbolic manifolds? How arethey different?
Strictly Convex⇒ very similar. Properly convex⇒ lesssimilar
• What sort of structure do convex projective manifoldshave?
Deformations of finite volume strictly convex manifolds arestructurally similar to complete finite volume hyperbolicmanifolds
![Page 4: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/4.jpg)
Some Questions
• What are convex projective manifolds?
Generalizations of Hyperbolic manifolds
• How are they similar to hyperbolic manifolds? How arethey different?
Strictly Convex⇒ very similar. Properly convex⇒ lesssimilar
• What sort of structure do convex projective manifoldshave?
Deformations of finite volume strictly convex manifolds arestructurally similar to complete finite volume hyperbolicmanifolds
![Page 5: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/5.jpg)
Some Questions
• What are convex projective manifolds?Generalizations of Hyperbolic manifolds
• How are they similar to hyperbolic manifolds? How arethey different?
Strictly Convex⇒ very similar. Properly convex⇒ lesssimilar
• What sort of structure do convex projective manifoldshave?
Deformations of finite volume strictly convex manifolds arestructurally similar to complete finite volume hyperbolicmanifolds
![Page 6: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/6.jpg)
Some Questions
• What are convex projective manifolds?Generalizations of Hyperbolic manifolds
• How are they similar to hyperbolic manifolds? How arethey different?Strictly Convex⇒ very similar. Properly convex⇒ lesssimilar
• What sort of structure do convex projective manifoldshave?
Deformations of finite volume strictly convex manifolds arestructurally similar to complete finite volume hyperbolicmanifolds
![Page 7: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/7.jpg)
Some Questions
• What are convex projective manifolds?Generalizations of Hyperbolic manifolds
• How are they similar to hyperbolic manifolds? How arethey different?Strictly Convex⇒ very similar. Properly convex⇒ lesssimilar
• What sort of structure do convex projective manifoldshave?Deformations of finite volume strictly convex manifolds arestructurally similar to complete finite volume hyperbolicmanifolds
![Page 8: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/8.jpg)
Projective Space
• RPn is the space of lines through origin in Rn+1.
• Let P : Rn+1\0 → RPn be the obvious projection.• The automorphism group of RPn is
PGLn+1(R) := GLn+1(R)/R×.• A codimension k projective plane is the projectivization of
a codimension k plane in Rn+1
• A projective line is the projectivization of a 2-plane in Rn+1
• A projective hyperplane is the projectivization of an n-planein Rn+1.
![Page 9: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/9.jpg)
Projective Space
• RPn is the space of lines through origin in Rn+1.• Let P : Rn+1\0 → RPn be the obvious projection.
• The automorphism group of RPn isPGLn+1(R) := GLn+1(R)/R×.
• A codimension k projective plane is the projectivization ofa codimension k plane in Rn+1
• A projective line is the projectivization of a 2-plane in Rn+1
• A projective hyperplane is the projectivization of an n-planein Rn+1.
![Page 10: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/10.jpg)
Projective Space
• RPn is the space of lines through origin in Rn+1.• Let P : Rn+1\0 → RPn be the obvious projection.• The automorphism group of RPn is
PGLn+1(R) := GLn+1(R)/R×.
• A codimension k projective plane is the projectivization ofa codimension k plane in Rn+1
• A projective line is the projectivization of a 2-plane in Rn+1
• A projective hyperplane is the projectivization of an n-planein Rn+1.
![Page 11: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/11.jpg)
Projective Space
• RPn is the space of lines through origin in Rn+1.• Let P : Rn+1\0 → RPn be the obvious projection.• The automorphism group of RPn is
PGLn+1(R) := GLn+1(R)/R×.• A codimension k projective plane is the projectivization of
a codimension k plane in Rn+1
• A projective line is the projectivization of a 2-plane in Rn+1
• A projective hyperplane is the projectivization of an n-planein Rn+1.
![Page 12: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/12.jpg)
Projective Space
• RPn is the space of lines through origin in Rn+1.• Let P : Rn+1\0 → RPn be the obvious projection.• The automorphism group of RPn is
PGLn+1(R) := GLn+1(R)/R×.• A codimension k projective plane is the projectivization of
a codimension k plane in Rn+1
• A projective line is the projectivization of a 2-plane in Rn+1
• A projective hyperplane is the projectivization of an n-planein Rn+1.
![Page 13: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/13.jpg)
Projective Space
• RPn is the space of lines through origin in Rn+1.• Let P : Rn+1\0 → RPn be the obvious projection.• The automorphism group of RPn is
PGLn+1(R) := GLn+1(R)/R×.• A codimension k projective plane is the projectivization of
a codimension k plane in Rn+1
• A projective line is the projectivization of a 2-plane in Rn+1
• A projective hyperplane is the projectivization of an n-planein Rn+1.
![Page 14: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/14.jpg)
A Decomposition of RPn
• Let H be a hyperplane in Rn+1.• H gives rise to a Decomposition of RPn = Rn t RPn−1 into
an affine part and an ideal part.
• RPn\P(H) is called an affine patch.
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A Decomposition of RPn
• Let H be a hyperplane in Rn+1.• H gives rise to a Decomposition of RPn = Rn t RPn−1 into
an affine part and an ideal part.
• RPn\P(H) is called an affine patch.
![Page 16: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/16.jpg)
A Decomposition of RPn
• Let H be a hyperplane in Rn+1.• H gives rise to a Decomposition of RPn = Rn t RPn−1 into
an affine part and an ideal part.
• RPn\P(H) is called an affine patch.
![Page 17: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/17.jpg)
What is convex projective geometry?Motivation from hyperbolic geometry
• Let 〈x , y〉 = x1y1 + . . . xnyn − xn+1yn+1 be the standardbilinear form of signature (n,1) on Rn+1
• Let C = x ∈ Rn+1|〈x , x〉 < 0
• P(C) is the Klein model of hyperbolic space.• P(C) has isometry group PSO(n,1) ≤ PGLn+1(R)
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What is convex projective geometry?Motivation from hyperbolic geometry
• Let 〈x , y〉 = x1y1 + . . . xnyn − xn+1yn+1 be the standardbilinear form of signature (n,1) on Rn+1
• Let C = x ∈ Rn+1|〈x , x〉 < 0• P(C) is the Klein model of hyperbolic space.• P(C) has isometry group PSO(n,1) ≤ PGLn+1(R)
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What is convex projective geometry?Motivation from hyperbolic geometry
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.
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What is convex projective geometry?Motivation from hyperbolic geometry
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.
![Page 21: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/21.jpg)
What is convex projective geometry?Motivation from hyperbolic geometry
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.
![Page 22: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/22.jpg)
What is convex projective geometry?Motivation from hyperbolic geometry
Convex projective geometry focuses on the geometry ofmanifolds that are locally modeled on properly (strictly) convexdomains.
Hyperbolic Geometry
Hn/Γ
Γ ≤ Isom(Hn)
Γ discrete + torsion free
Convex ProjectiveGeometry
Ω/Γ
Ω properly (strictly) convex
Γ ≤ PGL(Ω)
Γ discrete + torsion free
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What is convex projective geometry?Motivation from hyperbolic geometry
Convex projective geometry focuses on the geometry ofmanifolds that are locally modeled on properly (strictly) convexdomains.
Hyperbolic Geometry
Hn/Γ
Γ ≤ Isom(Hn)
Γ discrete + torsion free
Convex ProjectiveGeometry
Ω/Γ
Ω properly (strictly) convex
Γ ≤ PGL(Ω)
Γ discrete + torsion free
![Page 24: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/24.jpg)
What is convex projective geometry?Motivation from hyperbolic geometry
Convex projective geometry focuses on the geometry ofmanifolds that are locally modeled on properly (strictly) convexdomains.
Hyperbolic Geometry
Hn/Γ
Γ ≤ Isom(Hn)
Γ discrete + torsion free
Convex ProjectiveGeometry
Ω/Γ
Ω properly (strictly) convex
Γ ≤ PGL(Ω)
Γ discrete + torsion free
![Page 25: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/25.jpg)
What is Convex Projective GeometryExamples
1. Hyperbolic manifolds
2. Let T be the interior of a triangle in RP2 and let Γ ≤ Diag+
be a suitable lattice inside the group of 3× 3 diagonalmatrices with determinant 1 and distinct positiveeigenvalues. T/Γ is a properly convex torus.
These are extreme examples of properly convex manifolds.Generic examples interpolate between these extreme cases.
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What is Convex Projective GeometryExamples
1. Hyperbolic manifolds2. Let T be the interior of a triangle in RP2 and let Γ ≤ Diag+
be a suitable lattice inside the group of 3× 3 diagonalmatrices with determinant 1 and distinct positiveeigenvalues. T/Γ is a properly convex torus.
These are extreme examples of properly convex manifolds.Generic examples interpolate between these extreme cases.
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What is Convex Projective GeometryExamples
1. Hyperbolic manifolds2. Let T be the interior of a triangle in RP2 and let Γ ≤ Diag+
be a suitable lattice inside the group of 3× 3 diagonalmatrices with determinant 1 and distinct positiveeigenvalues. T/Γ is a properly convex torus.
These are extreme examples of properly convex manifolds.Generic examples interpolate between these extreme cases.
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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.
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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 30: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/30.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 31: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/31.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 32: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/32.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 33: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/33.jpg)
Classification of Isometriesa la Cooper, Long, Tillmann
If Ω is open and 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 (zero translation length +
realized),2. parabolic if γ acts freely on Ω and has all eigenvalues of
modulus 1 (zero translation length + not realized), and3. hyperbolic otherwise (positive translation length)
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Classification of Isometriesa la Cooper, Long, Tillmann
If Ω is open and 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 (zero translation length +realized),
2. parabolic if γ acts freely on Ω and has all eigenvalues ofmodulus 1 (zero translation length + not realized), and
3. hyperbolic otherwise (positive translation length)
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Similarities to Hyperbolic IsometriesStrictly Convex Case
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 IsometriesStrictly Convex Case
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 37: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/37.jpg)
Similarities to Hyperbolic IsometriesStrictly Convex Case
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 38: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/38.jpg)
Similarities to Hyperbolic IsometriesStrictly Convex Case
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 IsometriesStrictly Convex Case
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 IsometriesThe General Case
A properly convex domain is a compact convex subset of Rn
and so if γ ∈ PGL(Ω) then Brouwer fixed point theorem applies
• Elliptic elements are all conjugate into O(n).• Parabolic elements have a connected fixed set in ∂Ω.• Hyperbolic elements have an attracting and repelling
subspaces A+ and A− in ∂Ω. The action on these sets isorthogonal and their dimension is determined by thenumber of “powerful” Jordan blocks of γ
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Similarities to Hyperbolic IsometriesThe General Case
A properly convex domain is a compact convex subset of Rn
and so if γ ∈ PGL(Ω) then Brouwer fixed point theorem applies• Elliptic elements are all conjugate into O(n).
• Parabolic elements have a connected fixed set in ∂Ω.• Hyperbolic elements have an attracting and repelling
subspaces A+ and A− in ∂Ω. The action on these sets isorthogonal and their dimension is determined by thenumber of “powerful” Jordan blocks of γ
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Similarities to Hyperbolic IsometriesThe General Case
A properly convex domain is a compact convex subset of Rn
and so if γ ∈ PGL(Ω) then Brouwer fixed point theorem applies• Elliptic elements are all conjugate into O(n).• Parabolic elements have a connected fixed set in ∂Ω.
• Hyperbolic elements have an attracting and repellingsubspaces A+ and A− in ∂Ω. The action on these sets isorthogonal and their dimension is determined by thenumber of “powerful” Jordan blocks of γ
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Similarities to Hyperbolic IsometriesThe General Case
A properly convex domain is a compact convex subset of Rn
and so if γ ∈ PGL(Ω) then Brouwer fixed point theorem applies• Elliptic elements are all conjugate into O(n).• Parabolic elements have a connected fixed set in ∂Ω.• Hyperbolic elements have an attracting and repelling
subspaces A+ and A− in ∂Ω. The action on these sets isorthogonal and their dimension is determined by thenumber of “powerful” Jordan blocks of γ
![Page 44: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/44.jpg)
Margulis Lemma
Let Ω ⊂ RPn is an open properly convex domain and letΓ ≤ PGL(Ω) be a discrete group. Then there exists a numberµn (depending only on n) such that if x ∈ Ω then the group
Γx = 〈γ ∈ Γ|dΩ(x , γx) < µn〉
is virtually nilpotent.
• Γx can be thought of as the subgroup of Γ generated byloops in Ω/Γ of length at most µn passing through [x ].
• The Margulis lemma places restrictions on the topologyand geometry of the “thin” part of Ω/Γ.
Result due to Gromov-Margulis-Thurston for Hn andCooper-Long-Tillmann in general.
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Margulis Lemma
Let Ω ⊂ RPn is an open properly convex domain and letΓ ≤ PGL(Ω) be a discrete group. Then there exists a numberµn (depending only on n) such that if x ∈ Ω then the group
Γx = 〈γ ∈ Γ|dΩ(x , γx) < µn〉
is virtually nilpotent.
• Γx can be thought of as the subgroup of Γ generated byloops in Ω/Γ of length at most µn passing through [x ].
• The Margulis lemma places restrictions on the topologyand geometry of the “thin” part of Ω/Γ.
Result due to Gromov-Margulis-Thurston for Hn andCooper-Long-Tillmann in general.
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Margulis Lemma
Let Ω ⊂ RPn is an open properly convex domain and letΓ ≤ PGL(Ω) be a discrete group. Then there exists a numberµn (depending only on n) such that if x ∈ Ω then the group
Γx = 〈γ ∈ Γ|dΩ(x , γx) < µn〉
is virtually nilpotent.
• Γx can be thought of as the subgroup of Γ generated byloops in Ω/Γ of length at most µn passing through [x ].
• The Margulis lemma places restrictions on the topologyand geometry of the “thin” part of Ω/Γ.
Result due to Gromov-Margulis-Thurston for Hn andCooper-Long-Tillmann in general.
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Margulis Lemma
Let Ω ⊂ RPn is an open properly convex domain and letΓ ≤ PGL(Ω) be a discrete group. Then there exists a numberµn (depending only on n) such that if x ∈ Ω then the group
Γx = 〈γ ∈ Γ|dΩ(x , γx) < µn〉
is virtually nilpotent.
• Γx can be thought of as the subgroup of Γ generated byloops in Ω/Γ of length at most µn passing through [x ].
• The Margulis lemma places restrictions on the topologyand geometry of the “thin” part of Ω/Γ.
Result due to Gromov-Margulis-Thurston for Hn andCooper-Long-Tillmann in general.
![Page 48: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/48.jpg)
Rigidity and Flexibility
When n ≥ 3 Mostow-Prasad rigidity tells us that complete finitevolume hyperbolic structures are very rigid
Theorem 1 (Mostow ’70, Prasad ’73)Let n ≥ 3 and suppose that Hn/Γ1 and Hn/Γ2 both have finitevolume. If Γ1 and Γ2 are isomorphic then Hn/Γ1 and Hn/Γ2 areisometric.
There is no Mostow-Prasad rigidity for properly (strictly) convexdomains.There are examples of finite volume hyperbolic manifoldswhose complete hyperbolic structure can be “deformed” to anon-hyperbolic convex projective structure.
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Rigidity and Flexibility
When n ≥ 3 Mostow-Prasad rigidity tells us that complete finitevolume hyperbolic structures are very rigid
Theorem 1 (Mostow ’70, Prasad ’73)Let n ≥ 3 and suppose that Hn/Γ1 and Hn/Γ2 both have finitevolume. If Γ1 and Γ2 are isomorphic then Hn/Γ1 and Hn/Γ2 areisometric.There is no Mostow-Prasad rigidity for properly (strictly) convexdomains.There are examples of finite volume hyperbolic manifoldswhose complete hyperbolic structure can be “deformed” to anon-hyperbolic convex projective structure.
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Deformations
• Start with M0 = Ω0/Γ0 which is properly convex.
• “Perturb” Γ0 to Γ1 ≤ PGL(Ω1) ≤ PGLn+1(R), where Γ0∼= Γ1
and Ω1 is properly convex.• We say that M1 = Ω1/Γ1 is a deformation of M0
Ex: Let Ω0∼= Hn, Γ0 ≤ PSO(n,1), such that Ω0/Γ0 is finite
volume and contains an embedded totally geodesichypersurface Σ. Let Γ1 be obtained by “bending” along Σ.
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Deformations
• Start with M0 = Ω0/Γ0 which is properly convex.• “Perturb” Γ0 to Γ1 ≤ PGL(Ω1) ≤ PGLn+1(R), where Γ0
∼= Γ1and Ω1 is properly convex.
• We say that M1 = Ω1/Γ1 is a deformation of M0
Ex: Let Ω0∼= Hn, Γ0 ≤ PSO(n,1), such that Ω0/Γ0 is finite
volume and contains an embedded totally geodesichypersurface Σ. Let Γ1 be obtained by “bending” along Σ.
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Deformations
• Start with M0 = Ω0/Γ0 which is properly convex.• “Perturb” Γ0 to Γ1 ≤ PGL(Ω1) ≤ PGLn+1(R), where Γ0
∼= Γ1and Ω1 is properly convex.
• We say that M1 = Ω1/Γ1 is a deformation of M0
Ex: Let Ω0∼= Hn, Γ0 ≤ PSO(n,1), such that Ω0/Γ0 is finite
volume and contains an embedded totally geodesichypersurface Σ. Let Γ1 be obtained by “bending” along Σ.
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Deformations
• Start with M0 = Ω0/Γ0 which is properly convex.• “Perturb” Γ0 to Γ1 ≤ PGL(Ω1) ≤ PGLn+1(R), where Γ0
∼= Γ1and Ω1 is properly convex.
• We say that M1 = Ω1/Γ1 is a deformation of M0
Ex: Let Ω0∼= Hn, Γ0 ≤ PSO(n,1), such that Ω0/Γ0 is finite
volume and contains an embedded totally geodesichypersurface Σ. Let Γ1 be obtained by “bending” along Σ.
![Page 54: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/54.jpg)
Structure of Hyperbolic ManifoldsThe Closed Case
Let Hn/Γ be a closed hyperbolic manifold.• Since Γ acts cocompactly by isometries on Hn we see that
Γ is δ-hyperbolic group (Švarc-Milnor)
• By compactness, we see that if 1 6= γ ∈ Γ then γ ishyperbolic
• In particular, if 1 6= γ ∈ Γ then γ is positive proximal(eigenvalues of minimum and maximum modulus areunique, real, and positive)
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Structure of Hyperbolic ManifoldsThe Closed Case
Let Hn/Γ be a closed hyperbolic manifold.• Since Γ acts cocompactly by isometries on Hn we see that
Γ is δ-hyperbolic group (Švarc-Milnor)• By compactness, we see that if 1 6= γ ∈ Γ then γ is
hyperbolic
• In particular, if 1 6= γ ∈ Γ then γ is positive proximal(eigenvalues of minimum and maximum modulus areunique, real, and positive)
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Structure of Hyperbolic ManifoldsThe Closed Case
Let Hn/Γ be a closed hyperbolic manifold.• Since Γ acts cocompactly by isometries on Hn we see that
Γ is δ-hyperbolic group (Švarc-Milnor)• By compactness, we see that if 1 6= γ ∈ Γ then γ is
hyperbolic• In particular, if 1 6= γ ∈ Γ then γ is positive proximal
(eigenvalues of minimum and maximum modulus areunique, real, and positive)
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Structure of Convex Projective ManifoldsThe Closed Case
Let M = Ω/Γ be a closed properly convex manifold that is adeformation of a closed strictly convex manifold M0 = Ω0/Γ0.
Theorem 2 (Benoist)Suppose Ω/Γ is closed. Ω/Γ is strictly convex if and only if Γ isδ-hyperbolic.
⇐= Proof sketch.If Ω is not strictly convex then it willcontain arbitrarily fat triangles and isthus not δ-hyperbolic. Since Γ actscocompactly by isometries on Ω,Švarc-Milnor tells us that Ω is q.i. to Γand is thus δ-hyperbolic.
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Structure of Convex Projective ManifoldsThe Closed Case
Let M = Ω/Γ be a closed properly convex manifold that is adeformation of a closed strictly convex manifold M0 = Ω0/Γ0.
Theorem 2 (Benoist)Suppose Ω/Γ is closed. Ω/Γ is strictly convex if and only if Γ isδ-hyperbolic.
⇐= Proof sketch.If Ω is not strictly convex then it willcontain arbitrarily fat triangles and isthus not δ-hyperbolic. Since Γ actscocompactly by isometries on Ω,Švarc-Milnor tells us that Ω is q.i. to Γand is thus δ-hyperbolic.
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Structure of Convex Projective ManifoldsThe Closed Case
Let M = Ω/Γ be a closed properly convex manifold that is adeformation of a closed strictly convex manifold M0 = Ω0/Γ0.
Theorem 2 (Benoist)Suppose Ω/Γ is closed. Ω/Γ is strictly convex if and only if Γ isδ-hyperbolic.
⇐= Proof sketch.If Ω is not strictly convex then it willcontain arbitrarily fat triangles and isthus not δ-hyperbolic.
Since Γ actscocompactly by isometries on Ω,Švarc-Milnor tells us that Ω is q.i. to Γand is thus δ-hyperbolic.
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Structure of Convex Projective ManifoldsThe Closed Case
Let M = Ω/Γ be a closed properly convex manifold that is adeformation of a closed strictly convex manifold M0 = Ω0/Γ0.
Theorem 2 (Benoist)Suppose Ω/Γ is closed. Ω/Γ is strictly convex if and only if Γ isδ-hyperbolic.
⇐= Proof sketch.If Ω is not strictly convex then it willcontain arbitrarily fat triangles and isthus not δ-hyperbolic. Since Γ actscocompactly by isometries on Ω,Švarc-Milnor tells us that Ω is q.i. to Γand is thus δ-hyperbolic.
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Structure of Convex Projective ManifoldsThe Closed Case
Theorem 3 (Benoist)Let 1 6= γ ∈ Γ then γ is positive proximal.
Proof.
• Again by compactness we have that if 1 6= γ ∈ Γ then γ ishyperbolic.
• Since Ω is strictly convex and γ is hyperbolic we see that γhas exactly 2 fixed points in ∂Ω and acts as translationalong the geodesic connecting them. γ is thus positiveproximal.
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Structure of Convex Projective ManifoldsThe Closed Case
Theorem 3 (Benoist)Let 1 6= γ ∈ Γ then γ is positive proximal.
Proof.
• Again by compactness we have that if 1 6= γ ∈ Γ then γ ishyperbolic.
• Since Ω is strictly convex and γ is hyperbolic we see that γhas exactly 2 fixed points in ∂Ω and acts as translationalong the geodesic connecting them. γ is thus positiveproximal.
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Structure of Hyperbolic ManifoldsFinite Volume Case
Let M = Hn/Γ be a finite volume hyperbolic manifold. We candecompose M as
M = MK⊔
i
Ci ,
where MK is a compact and π1(MK ) = Γ and Ci arecomponents of the thin part called cusps.
As we will see, the Margulis lemma tells us that the Ci haverelatively simple geometry.
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Structure of Hyperbolic ManifoldsFinite Volume Case
Let M = Hn/Γ be a finite volume hyperbolic manifold. We candecompose M as
M = MK⊔
i
Ci ,
where MK is a compact and π1(MK ) = Γ and Ci arecomponents of the thin part called cusps.As we will see, the Margulis lemma tells us that the Ci haverelatively simple geometry.
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Geometry of the Cusps
Let C be a cusp of a finite volume hyperbolic manifold and let
P =
1 vT |v |2
0 In−1 v0 0 1
|v ∈ Rn−1
be the group of parabolic translations fixing∞. Let x0 ∈ Hn,then C ∼= B/∆ where B is horoball bounded by Px0 and ∆ is afinite extension of a lattice in P.
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Structure of Hyperbolic ManifoldsThe Finite Volume Case
• Γ no longer acts cocompactly on Hn and Γ is no longerδ-hyperbolic
• Instead Γ is δ-hyperbolic relative to the cusps• If 1 6= γ ∈ Γ is freely homotopic into a cusp then γ is
parabolic, otherwise γ is hyperbolic (positive proximal)
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Structure of Hyperbolic ManifoldsThe Finite Volume Case
• Γ no longer acts cocompactly on Hn and Γ is no longerδ-hyperbolic
• Instead Γ is δ-hyperbolic relative to the cusps
• If 1 6= γ ∈ Γ is freely homotopic into a cusp then γ isparabolic, otherwise γ is hyperbolic (positive proximal)
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Structure of Hyperbolic ManifoldsThe Finite Volume Case
• Γ no longer acts cocompactly on Hn and Γ is no longerδ-hyperbolic
• Instead Γ is δ-hyperbolic relative to the cusps• If 1 6= γ ∈ Γ is freely homotopic into a cusp then γ is
parabolic, otherwise γ is hyperbolic (positive proximal)
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Structures of Convex Projective ManifoldsThe Strictly Convex Finite Volume Case
Let Ω/Γ be a finite volume (Hausdorff measure of Hilbertmetric) strictly convex manifold.
Theorem 4 (Cooper, Long, Tillmann ‘11)Let M = Ω/Γ be as above then• M = MK
⊔i Ci , where MK is compact and Ci is projectively
equivalent to the cusp of a finite volume hyperbolicmanifold,
• Γ is δ-hyperbolic relative to its cusps, and• If 1 6= γ ∈ Γ is freely homotopic into a cusp then γ is
parabolic. Otherwise γ is hyperbolic (positive proximal).
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Figure-8 ExampleConsider the following example.
Let K be the figure-8 knot, let M = S3\K , and let G = π1(M)
Theorem 5 (B)There exists ε > 0 such that for each t ∈ (−ε, ε) there is aproperly convex domain Ωt and a discrete group Γt ≤ PGL(Ωt )such that• Ωt/Γt ∼= M,• Ω0/Γ0 is the complete hyperbolic structure on M, and• If t 6= 0 then Ωt is not strictly convex.
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Figure-8 ExampleConsider the following example.
Let K be the figure-8 knot, let M = S3\K , and let G = π1(M)
Theorem 5 (B)There exists ε > 0 such that for each t ∈ (−ε, ε) there is aproperly convex domain Ωt and a discrete group Γt ≤ PGL(Ωt )such that• Ωt/Γt ∼= M,• Ω0/Γ0 is the complete hyperbolic structure on M, and• If t 6= 0 then Ωt is not strictly convex.
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Figure-8 Example
Theorem 6 (B)For each t ∈ (−ε, ε) we can decompose Ωt/Γt as M t
K⊔
Ct ,where M t
K is compact and Ct ∼= T 2 × [1,∞).
• For each t , Ct ∼= Bt/∆t , where ∆t is a lattice an Abeliangroup Pt of “translations,” and Bt is a “horoball” bounded byan orbit of Pt .
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Figure-8 Example
• For each t 6= 0 there is 1 6= γt ∈ Γt such that γt ishyperbolic, freely homotopic into Ct , but not positiveproximal.
• Ωt contains non-trivial line segments in ∂Ωt that arepreserved by conjugates of ∆t . In particular, Ωt is notδ-hyperbolic.
![Page 74: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/74.jpg)
Figure-8 Example
• For each t 6= 0 there is 1 6= γt ∈ Γt such that γt ishyperbolic, freely homotopic into Ct , but not positiveproximal.
• Ωt contains non-trivial line segments in ∂Ωt that arepreserved by conjugates of ∆t . In particular, Ωt is notδ-hyperbolic.
![Page 75: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/75.jpg)
Figure-8 Example
• For each t 6= 0 there is 1 6= γt ∈ Γt such that γt ishyperbolic, freely homotopic into Ct , but not positiveproximal.
• Ωt contains non-trivial line segments in ∂Ωt that arepreserved by conjugates of ∆t . In particular, Ωt is notδ-hyperbolic.
![Page 76: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/76.jpg)
Figure-8 Example
• For each t 6= 0 there is 1 6= γt ∈ Γt such that γt ishyperbolic, freely homotopic into Ct , but not positiveproximal.
• Ωt contains non-trivial line segments in ∂Ωt that arepreserved by conjugates of ∆t . In particular, Ωt is notδ-hyperbolic.
![Page 77: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/77.jpg)
Figure-8 Example
• For each t 6= 0 there is 1 6= γt ∈ Γt such that γt ishyperbolic, freely homotopic into Ct , but not positiveproximal.
• Ωt contains non-trivial line segments in ∂Ωt that arepreserved by conjugates of ∆t . In particular, Ωt is notδ-hyperbolic.
![Page 78: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/78.jpg)
Figure-8 Example
• For each t 6= 0 there is 1 6= γt ∈ Γt such that γt ishyperbolic, freely homotopic into Ct , but not positiveproximal.
• Ωt contains non-trivial line segments in ∂Ωt that arepreserved by conjugates of ∆t . In particular, Ωt is notδ-hyperbolic.
![Page 79: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/79.jpg)
Figure-8 Example
Theorem 7 (B, Long)1 6= γ ∈ Γt is positive proximal if and only if it cannot be freelyhomotoped into Ct .
Proof.⇐ Let 1 6= γ ∈ Γt . No elements of Pt are positive proximal, so ifγ is freely homotopic to Ct then it is not positive proximal.
⇒ If γ is not freely homotopic to Ct then γ has positivetranslation length and is thus hyperbolic. Furthermore, thistranslation length is realized by points on an axis.
![Page 80: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/80.jpg)
Figure-8 Example
Theorem 7 (B, Long)1 6= γ ∈ Γt is positive proximal if and only if it cannot be freelyhomotoped into Ct .
Proof.⇐ Let 1 6= γ ∈ Γt . No elements of Pt are positive proximal, so ifγ is freely homotopic to Ct then it is not positive proximal.
⇒ If γ is not freely homotopic to Ct then γ has positivetranslation length and is thus hyperbolic. Furthermore, thistranslation length is realized by points on an axis.
![Page 81: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/81.jpg)
Figure-8 Example
Theorem 7 (B, Long)1 6= γ ∈ Γt is positive proximal if and only if it cannot be freelyhomotoped into Ct .
Proof.⇐ Let 1 6= γ ∈ Γt . No elements of Pt are positive proximal, so ifγ is freely homotopic to Ct then it is not positive proximal.
⇒ If γ is not freely homotopic to Ct then γ has positivetranslation length and is thus hyperbolic. Furthermore, thistranslation length is realized by points on an axis.
![Page 82: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/82.jpg)
Figure-8 Example
Proof (Continued).Use Margulis lemma to construct a disjoint and Γt invariantcollection Ht of horoballs in Ωt .
let Ωt be the electric space obtained by collapsing thehorospherical boundary components of Ωt\Ht .
Lemma 8 (B, Long)Ωt is δ-hyperbolic
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Figure-8 Example
Proof (Continued).Use Margulis lemma to construct a disjoint and Γt invariantcollection Ht of horoballs in Ωt .let Ωt be the electric space obtained by collapsing thehorospherical boundary components of Ωt\Ht .
Lemma 8 (B, Long)Ωt is δ-hyperbolic
![Page 84: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/84.jpg)
Figure-8 Example
Proof (Continued).Use Margulis lemma to construct a disjoint and Γt invariantcollection Ht of horoballs in Ωt .let Ωt be the electric space obtained by collapsing thehorospherical boundary components of Ωt\Ht .
Lemma 8 (B, Long)Ωt is δ-hyperbolic
![Page 85: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/85.jpg)
Figure-8 ExampleProof (Continued).
• Since γ is hyperbolic and preserves Ωt we know that γ hasreal eigenvalues of largest and smallest modulus and thatthese eigenvalues have the same sign.
• If γ is not positive proximal then there will be a γ-invariantset T ⊂ Ωt disjoint from all the horoballs that contains apositive dimensional flat in its boundary
• This gives rise to arbitrarily fat triangles in Ωt
![Page 86: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/86.jpg)
Figure-8 ExampleProof (Continued).
• Since γ is hyperbolic and preserves Ωt we know that γ hasreal eigenvalues of largest and smallest modulus and thatthese eigenvalues have the same sign.
• If γ is not positive proximal then there will be a γ-invariantset T ⊂ Ωt disjoint from all the horoballs that contains apositive dimensional flat in its boundary
• This gives rise to arbitrarily fat triangles in Ωt
![Page 87: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/87.jpg)
Figure-8 ExampleProof (Continued).
• Since γ is hyperbolic and preserves Ωt we know that γ hasreal eigenvalues of largest and smallest modulus and thatthese eigenvalues have the same sign.
• If γ is not positive proximal then there will be a γ-invariantset T ⊂ Ωt disjoint from all the horoballs that contains apositive dimensional flat in its boundary
• This gives rise to arbitrarily fat triangles in Ωt
![Page 88: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/88.jpg)
Figure-8 ExampleProof (Continued).
• Since γ is hyperbolic and preserves Ωt we know that γ hasreal eigenvalues of largest and smallest modulus and thatthese eigenvalues have the same sign.
• If γ is not positive proximal then there will be a γ-invariantset T ⊂ Ωt disjoint from all the horoballs that contains apositive dimensional flat in its boundary
• This gives rise to arbitrarily fat triangles in Ωt
![Page 89: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/89.jpg)
Summary and Questions
• The structure of a finite volume strictly convex manifold iswell understood.
• As you deform the structure the “coarse” geometry of thecompact part doesn’t change.
• The geometry of the cusps may change as we deform, butcan be understood using the Margulis lemma.
• Theorem 7 holds for all properly convex deformations offinite volume strictly convex manifolds in dimension 3
• Theorem 7 should hold for higher dimensions.• What can we say for deformations of deformations of
infinite volume hyperbolic manifolds?
![Page 90: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/90.jpg)
Summary and Questions
• The structure of a finite volume strictly convex manifold iswell understood.
• As you deform the structure the “coarse” geometry of thecompact part doesn’t change.
• The geometry of the cusps may change as we deform, butcan be understood using the Margulis lemma.
• Theorem 7 holds for all properly convex deformations offinite volume strictly convex manifolds in dimension 3
• Theorem 7 should hold for higher dimensions.• What can we say for deformations of deformations of
infinite volume hyperbolic manifolds?
![Page 91: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/91.jpg)
Summary and Questions
• The structure of a finite volume strictly convex manifold iswell understood.
• As you deform the structure the “coarse” geometry of thecompact part doesn’t change.
• The geometry of the cusps may change as we deform, butcan be understood using the Margulis lemma.
• Theorem 7 holds for all properly convex deformations offinite volume strictly convex manifolds in dimension 3
• Theorem 7 should hold for higher dimensions.• What can we say for deformations of deformations of
infinite volume hyperbolic manifolds?
![Page 92: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/92.jpg)
Summary and Questions
• The structure of a finite volume strictly convex manifold iswell understood.
• As you deform the structure the “coarse” geometry of thecompact part doesn’t change.
• The geometry of the cusps may change as we deform, butcan be understood using the Margulis lemma.
• Theorem 7 holds for all properly convex deformations offinite volume strictly convex manifolds in dimension 3
• Theorem 7 should hold for higher dimensions.• What can we say for deformations of deformations of
infinite volume hyperbolic manifolds?
![Page 93: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/93.jpg)
Summary and Questions
• The structure of a finite volume strictly convex manifold iswell understood.
• As you deform the structure the “coarse” geometry of thecompact part doesn’t change.
• The geometry of the cusps may change as we deform, butcan be understood using the Margulis lemma.
• Theorem 7 holds for all properly convex deformations offinite volume strictly convex manifolds in dimension 3
• Theorem 7 should hold for higher dimensions.
• What can we say for deformations of deformations ofinfinite volume hyperbolic manifolds?
![Page 94: The Structure of Properly Convex Manifoldsweb.math.ucsb.edu/~sballas/research/documents/AustinRTGworkshop.pdfWhat is convex projective geometry? Motivation from hyperbolic geometry](https://reader033.fdocuments.us/reader033/viewer/2022050111/5f48ffb60ad3bb3a38544627/html5/thumbnails/94.jpg)
Summary and Questions
• The structure of a finite volume strictly convex manifold iswell understood.
• As you deform the structure the “coarse” geometry of thecompact part doesn’t change.
• The geometry of the cusps may change as we deform, butcan be understood using the Margulis lemma.
• Theorem 7 holds for all properly convex deformations offinite volume strictly convex manifolds in dimension 3
• Theorem 7 should hold for higher dimensions.• What can we say for deformations of deformations of
infinite volume hyperbolic manifolds?