deformations of geometric structures and representations ...wmg/kaist.pdf · geometry through...
TRANSCRIPT
![Page 1: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/1.jpg)
Deformations of geometric structures andrepresentations of fundamental groups
William M. Goldman
Department of Mathematics University of Maryland
Seventh KAIST Geometric Topology FairGyeongju, KoreaJuly 9-11, 2007
http://www.math.umd.edu/∼wmg/kaist.pdf
![Page 2: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/2.jpg)
Enhancing Topology with Geometry
Deformations of geometric structure
Real projective structures
Representation varieties and character varieties
Hamiltonian flows of real projective structures
![Page 3: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/3.jpg)
Geometry through symmetryI In his 1872 Erlangen Program, Felix Klein proposed that a
geometry is the study of properties of an abstract space X
which are invariant under a transitive group G oftransformations of X .
I Klein was heavily influenced by Sophus Lie, who was trying todevelop a theory of continuous groups, to exploit infinitesimal
symmetry to study differential equations, similar to howGalois exploited symmetry to study algebraic equations.
![Page 4: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/4.jpg)
Geometry through symmetryI In his 1872 Erlangen Program, Felix Klein proposed that a
geometry is the study of properties of an abstract space X
which are invariant under a transitive group G oftransformations of X .
I Klein was heavily influenced by Sophus Lie, who was trying todevelop a theory of continuous groups, to exploit infinitesimal
symmetry to study differential equations, similar to howGalois exploited symmetry to study algebraic equations.
![Page 5: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/5.jpg)
Geometry through symmetryI In his 1872 Erlangen Program, Felix Klein proposed that a
geometry is the study of properties of an abstract space X
which are invariant under a transitive group G oftransformations of X .
I Klein was heavily influenced by Sophus Lie, who was trying todevelop a theory of continuous groups, to exploit infinitesimal
symmetry to study differential equations, similar to howGalois exploited symmetry to study algebraic equations.
![Page 6: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/6.jpg)
Euclidean to affine to projective geometry
I Euclidean geometry: X = Rn Euclidean space andG = Isom(X ) the group of rigid motions:
I A rigid motion is a map x 7→ Ax + b where A ∈ O(n) isorthogonal and b ∈ Rn is a translation vector.
I Invariant notions: Distance, angle, parallel, area, lines, ...
I Euclidean geometry: special case of affine geometrywheree X = Rn and G = Aff(X ), where A ∈ GL(n,R) is onlyrequired to be linear .
I Only parallelism, lines preserved.
I Affine geometry: special case of projective geometry, whenparallelism abandoned. G = PGL(n + 1,R), X = RPn.
I But the space must be enlarged: Rn $ RPn
![Page 7: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/7.jpg)
Euclidean to affine to projective geometry
I Euclidean geometry: X = Rn Euclidean space andG = Isom(X ) the group of rigid motions:
I A rigid motion is a map x 7→ Ax + b where A ∈ O(n) isorthogonal and b ∈ Rn is a translation vector.
I Invariant notions: Distance, angle, parallel, area, lines, ...
I Euclidean geometry: special case of affine geometrywheree X = Rn and G = Aff(X ), where A ∈ GL(n,R) is onlyrequired to be linear .
I Only parallelism, lines preserved.
I Affine geometry: special case of projective geometry, whenparallelism abandoned. G = PGL(n + 1,R), X = RPn.
I But the space must be enlarged: Rn $ RPn
![Page 8: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/8.jpg)
Euclidean to affine to projective geometry
I Euclidean geometry: X = Rn Euclidean space andG = Isom(X ) the group of rigid motions:
I A rigid motion is a map x 7→ Ax + b where A ∈ O(n) isorthogonal and b ∈ Rn is a translation vector.
I Invariant notions: Distance, angle, parallel, area, lines, ...
I Euclidean geometry: special case of affine geometrywheree X = Rn and G = Aff(X ), where A ∈ GL(n,R) is onlyrequired to be linear .
I Only parallelism, lines preserved.
I Affine geometry: special case of projective geometry, whenparallelism abandoned. G = PGL(n + 1,R), X = RPn.
I But the space must be enlarged: Rn $ RPn
![Page 9: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/9.jpg)
Euclidean to affine to projective geometry
I Euclidean geometry: X = Rn Euclidean space andG = Isom(X ) the group of rigid motions:
I A rigid motion is a map x 7→ Ax + b where A ∈ O(n) isorthogonal and b ∈ Rn is a translation vector.
I Invariant notions: Distance, angle, parallel, area, lines, ...
I Euclidean geometry: special case of affine geometrywheree X = Rn and G = Aff(X ), where A ∈ GL(n,R) is onlyrequired to be linear .
I Only parallelism, lines preserved.
I Affine geometry: special case of projective geometry, whenparallelism abandoned. G = PGL(n + 1,R), X = RPn.
I But the space must be enlarged: Rn $ RPn
![Page 10: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/10.jpg)
Euclidean to affine to projective geometry
I Euclidean geometry: X = Rn Euclidean space andG = Isom(X ) the group of rigid motions:
I A rigid motion is a map x 7→ Ax + b where A ∈ O(n) isorthogonal and b ∈ Rn is a translation vector.
I Invariant notions: Distance, angle, parallel, area, lines, ...
I Euclidean geometry: special case of affine geometrywheree X = Rn and G = Aff(X ), where A ∈ GL(n,R) is onlyrequired to be linear .
I Only parallelism, lines preserved.
I Affine geometry: special case of projective geometry, whenparallelism abandoned. G = PGL(n + 1,R), X = RPn.
I But the space must be enlarged: Rn $ RPn
![Page 11: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/11.jpg)
Euclidean to affine to projective geometry
I Euclidean geometry: X = Rn Euclidean space andG = Isom(X ) the group of rigid motions:
I A rigid motion is a map x 7→ Ax + b where A ∈ O(n) isorthogonal and b ∈ Rn is a translation vector.
I Invariant notions: Distance, angle, parallel, area, lines, ...
I Euclidean geometry: special case of affine geometrywheree X = Rn and G = Aff(X ), where A ∈ GL(n,R) is onlyrequired to be linear .
I Only parallelism, lines preserved.
I Affine geometry: special case of projective geometry, whenparallelism abandoned. G = PGL(n + 1,R), X = RPn.
I But the space must be enlarged: Rn $ RPn
![Page 12: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/12.jpg)
Euclidean to affine to projective geometry
I Euclidean geometry: X = Rn Euclidean space andG = Isom(X ) the group of rigid motions:
I A rigid motion is a map x 7→ Ax + b where A ∈ O(n) isorthogonal and b ∈ Rn is a translation vector.
I Invariant notions: Distance, angle, parallel, area, lines, ...
I Euclidean geometry: special case of affine geometrywheree X = Rn and G = Aff(X ), where A ∈ GL(n,R) is onlyrequired to be linear .
I Only parallelism, lines preserved.
I Affine geometry: special case of projective geometry, whenparallelism abandoned. G = PGL(n + 1,R), X = RPn.
I But the space must be enlarged: Rn $ RPn
![Page 13: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/13.jpg)
Euclidean to affine to projective geometry
I Euclidean geometry: X = Rn Euclidean space andG = Isom(X ) the group of rigid motions:
I A rigid motion is a map x 7→ Ax + b where A ∈ O(n) isorthogonal and b ∈ Rn is a translation vector.
I Invariant notions: Distance, angle, parallel, area, lines, ...
I Euclidean geometry: special case of affine geometrywheree X = Rn and G = Aff(X ), where A ∈ GL(n,R) is onlyrequired to be linear .
I Only parallelism, lines preserved.
I Affine geometry: special case of projective geometry, whenparallelism abandoned. G = PGL(n + 1,R), X = RPn.
I But the space must be enlarged: Rn $ RPn
![Page 14: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/14.jpg)
Other subgeometries of projective geometry
I Hyperbolic geometry: X = Hn ⊂ RPn G = O(n, 1) the subsetof PGL(n + 1,R) stabilizing X ;
I (Beltrami – Hilbert) Define the hyperbolic metric on X
projectively in terms of cross-ratios:
y
B
A
x
Distance d(x , y) = log[A, x , y ,B ]
I More generally, one obtains a projectively invariant distanceon any properly convex domain (Hilbert).
![Page 15: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/15.jpg)
Other subgeometries of projective geometry
I Hyperbolic geometry: X = Hn ⊂ RPn G = O(n, 1) the subsetof PGL(n + 1,R) stabilizing X ;
I (Beltrami – Hilbert) Define the hyperbolic metric on X
projectively in terms of cross-ratios:
y
B
A
x
Distance d(x , y) = log[A, x , y ,B ]
I More generally, one obtains a projectively invariant distanceon any properly convex domain (Hilbert).
![Page 16: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/16.jpg)
Other subgeometries of projective geometry
I Hyperbolic geometry: X = Hn ⊂ RPn G = O(n, 1) the subsetof PGL(n + 1,R) stabilizing X ;
I (Beltrami – Hilbert) Define the hyperbolic metric on X
projectively in terms of cross-ratios:
y
B
A
x
Distance d(x , y) = log[A, x , y ,B ]
I More generally, one obtains a projectively invariant distanceon any properly convex domain (Hilbert).
![Page 17: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/17.jpg)
Other subgeometries of projective geometry
I Hyperbolic geometry: X = Hn ⊂ RPn G = O(n, 1) the subsetof PGL(n + 1,R) stabilizing X ;
I (Beltrami – Hilbert) Define the hyperbolic metric on X
projectively in terms of cross-ratios:
y
B
A
x
Distance d(x , y) = log[A, x , y ,B ]
I More generally, one obtains a projectively invariant distanceon any properly convex domain (Hilbert).
![Page 18: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/18.jpg)
Putting geometric structure on a topological space
I Topology: Smooth manifold Σ with coordinate patches Uα;
I Charts — diffeomorphisms
Uαψα
−−→ ψα(Uα) ⊂ X
I For each component C ⊂ Uα ∩ Uβ, ∃g = g(C ) ∈ G such that
g ψα|C = ψβ |C .
I Local (G ,X )-geometry defined by ψα independent of patch.
I (Ehresmann 1936) Σ acquires geometric structure M modeled
on (G ,X ).
![Page 19: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/19.jpg)
Putting geometric structure on a topological space
I Topology: Smooth manifold Σ with coordinate patches Uα;
I Charts — diffeomorphisms
Uαψα
−−→ ψα(Uα) ⊂ X
I For each component C ⊂ Uα ∩ Uβ, ∃g = g(C ) ∈ G such that
g ψα|C = ψβ |C .
I Local (G ,X )-geometry defined by ψα independent of patch.
I (Ehresmann 1936) Σ acquires geometric structure M modeled
on (G ,X ).
![Page 20: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/20.jpg)
Putting geometric structure on a topological space
I Topology: Smooth manifold Σ with coordinate patches Uα;
I Charts — diffeomorphisms
Uαψα
−−→ ψα(Uα) ⊂ X
I For each component C ⊂ Uα ∩ Uβ, ∃g = g(C ) ∈ G such that
g ψα|C = ψβ |C .
I Local (G ,X )-geometry defined by ψα independent of patch.
I (Ehresmann 1936) Σ acquires geometric structure M modeled
on (G ,X ).
![Page 21: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/21.jpg)
Putting geometric structure on a topological space
I Topology: Smooth manifold Σ with coordinate patches Uα;
I Charts — diffeomorphisms
Uαψα
−−→ ψα(Uα) ⊂ X
I For each component C ⊂ Uα ∩ Uβ, ∃g = g(C ) ∈ G such that
g ψα|C = ψβ |C .
I Local (G ,X )-geometry defined by ψα independent of patch.
I (Ehresmann 1936) Σ acquires geometric structure M modeled
on (G ,X ).
![Page 22: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/22.jpg)
Putting geometric structure on a topological space
I Topology: Smooth manifold Σ with coordinate patches Uα;
I Charts — diffeomorphisms
Uαψα
−−→ ψα(Uα) ⊂ X
I For each component C ⊂ Uα ∩ Uβ, ∃g = g(C ) ∈ G such that
g ψα|C = ψβ |C .
I Local (G ,X )-geometry defined by ψα independent of patch.
I (Ehresmann 1936) Σ acquires geometric structure M modeled
on (G ,X ).
![Page 23: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/23.jpg)
Putting geometric structure on a topological space
I Topology: Smooth manifold Σ with coordinate patches Uα;
I Charts — diffeomorphisms
Uαψα
−−→ ψα(Uα) ⊂ X
I For each component C ⊂ Uα ∩ Uβ, ∃g = g(C ) ∈ G such that
g ψα|C = ψβ |C .
I Local (G ,X )-geometry defined by ψα independent of patch.
I (Ehresmann 1936) Σ acquires geometric structure M modeled
on (G ,X ).
![Page 24: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/24.jpg)
Quotients of domains
I Suppose that Ω ⊂ X is an open subset invariant under asubgroup Γ ⊂ G such that:
I Γ is discrete;I Γ acts properly on ΩI Γ acts freely on Ω.
I Then M = Ω/Γ is a (G ,X )-manifold
I The covering space Ω −→ M is a (G ,X )-morphism.
I Complete affine structures: Ω entire affine patch Rn ⊂ RPn.
I Convex RPn-structures: Ω ⊂ RPn convex domain containingno affine line (properly convex).
![Page 25: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/25.jpg)
Quotients of domains
I Suppose that Ω ⊂ X is an open subset invariant under asubgroup Γ ⊂ G such that:
I Γ is discrete;I Γ acts properly on ΩI Γ acts freely on Ω.
I Then M = Ω/Γ is a (G ,X )-manifold
I The covering space Ω −→ M is a (G ,X )-morphism.
I Complete affine structures: Ω entire affine patch Rn ⊂ RPn.
I Convex RPn-structures: Ω ⊂ RPn convex domain containingno affine line (properly convex).
![Page 26: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/26.jpg)
Quotients of domains
I Suppose that Ω ⊂ X is an open subset invariant under asubgroup Γ ⊂ G such that:
I Γ is discrete;I Γ acts properly on ΩI Γ acts freely on Ω.
I Then M = Ω/Γ is a (G ,X )-manifold
I The covering space Ω −→ M is a (G ,X )-morphism.
I Complete affine structures: Ω entire affine patch Rn ⊂ RPn.
I Convex RPn-structures: Ω ⊂ RPn convex domain containingno affine line (properly convex).
![Page 27: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/27.jpg)
Quotients of domains
I Suppose that Ω ⊂ X is an open subset invariant under asubgroup Γ ⊂ G such that:
I Γ is discrete;I Γ acts properly on ΩI Γ acts freely on Ω.
I Then M = Ω/Γ is a (G ,X )-manifold
I The covering space Ω −→ M is a (G ,X )-morphism.
I Complete affine structures: Ω entire affine patch Rn ⊂ RPn.
I Convex RPn-structures: Ω ⊂ RPn convex domain containingno affine line (properly convex).
![Page 28: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/28.jpg)
Quotients of domains
I Suppose that Ω ⊂ X is an open subset invariant under asubgroup Γ ⊂ G such that:
I Γ is discrete;I Γ acts properly on ΩI Γ acts freely on Ω.
I Then M = Ω/Γ is a (G ,X )-manifold
I The covering space Ω −→ M is a (G ,X )-morphism.
I Complete affine structures: Ω entire affine patch Rn ⊂ RPn.
I Convex RPn-structures: Ω ⊂ RPn convex domain containingno affine line (properly convex).
![Page 29: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/29.jpg)
Quotients of domains
I Suppose that Ω ⊂ X is an open subset invariant under asubgroup Γ ⊂ G such that:
I Γ is discrete;I Γ acts properly on ΩI Γ acts freely on Ω.
I Then M = Ω/Γ is a (G ,X )-manifold
I The covering space Ω −→ M is a (G ,X )-morphism.
I Complete affine structures: Ω entire affine patch Rn ⊂ RPn.
I Convex RPn-structures: Ω ⊂ RPn convex domain containingno affine line (properly convex).
![Page 30: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/30.jpg)
Quotients of domains
I Suppose that Ω ⊂ X is an open subset invariant under asubgroup Γ ⊂ G such that:
I Γ is discrete;I Γ acts properly on ΩI Γ acts freely on Ω.
I Then M = Ω/Γ is a (G ,X )-manifold
I The covering space Ω −→ M is a (G ,X )-morphism.
I Complete affine structures: Ω entire affine patch Rn ⊂ RPn.
I Convex RPn-structures: Ω ⊂ RPn convex domain containingno affine line (properly convex).
![Page 31: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/31.jpg)
Quotients of domains
I Suppose that Ω ⊂ X is an open subset invariant under asubgroup Γ ⊂ G such that:
I Γ is discrete;I Γ acts properly on ΩI Γ acts freely on Ω.
I Then M = Ω/Γ is a (G ,X )-manifold
I The covering space Ω −→ M is a (G ,X )-morphism.
I Complete affine structures: Ω entire affine patch Rn ⊂ RPn.
I Convex RPn-structures: Ω ⊂ RPn convex domain containingno affine line (properly convex).
![Page 32: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/32.jpg)
Quotients of domains
I Suppose that Ω ⊂ X is an open subset invariant under asubgroup Γ ⊂ G such that:
I Γ is discrete;I Γ acts properly on ΩI Γ acts freely on Ω.
I Then M = Ω/Γ is a (G ,X )-manifold
I The covering space Ω −→ M is a (G ,X )-morphism.
I Complete affine structures: Ω entire affine patch Rn ⊂ RPn.
I Convex RPn-structures: Ω ⊂ RPn convex domain containingno affine line (properly convex).
![Page 33: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/33.jpg)
A projective (3, 3, 3) triangle tesselation
This tesselation of the open triangular region is equivalent to thetiling of the Euclidean plane by equilateral triangles.
![Page 34: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/34.jpg)
Examples of incomplete quotient affine structures
![Page 35: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/35.jpg)
Hyperbolic structures as RP2-structures
I Using the Klein-Beltrami model of hyperbolic geometry, theconvex domain Ω bounded by a conic inherits a projectivelyinvariant hyperbolic geometry.
I The charts for the hyperbolic structure determine charts foran RP2-structure.
I Every hyperbolic manifold is convex RP2-manifold.
I A tiling of Ω = H2 in the projective model by triangles withangles π/3, π/3, π/4. The corresponding Coxeter groupcontains a finite index subgroup Γ such that Ω/Γ is a closedhyperbolic (and hence convex RP2-) surface.
![Page 36: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/36.jpg)
Hyperbolic structures as RP2-structures
I Using the Klein-Beltrami model of hyperbolic geometry, theconvex domain Ω bounded by a conic inherits a projectivelyinvariant hyperbolic geometry.
I The charts for the hyperbolic structure determine charts foran RP2-structure.
I Every hyperbolic manifold is convex RP2-manifold.
I A tiling of Ω = H2 in the projective model by triangles withangles π/3, π/3, π/4. The corresponding Coxeter groupcontains a finite index subgroup Γ such that Ω/Γ is a closedhyperbolic (and hence convex RP2-) surface.
![Page 37: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/37.jpg)
Hyperbolic structures as RP2-structures
I Using the Klein-Beltrami model of hyperbolic geometry, theconvex domain Ω bounded by a conic inherits a projectivelyinvariant hyperbolic geometry.
I The charts for the hyperbolic structure determine charts foran RP2-structure.
I Every hyperbolic manifold is convex RP2-manifold.
I A tiling of Ω = H2 in the projective model by triangles withangles π/3, π/3, π/4. The corresponding Coxeter groupcontains a finite index subgroup Γ such that Ω/Γ is a closedhyperbolic (and hence convex RP2-) surface.
![Page 38: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/38.jpg)
Hyperbolic structures as RP2-structures
I Using the Klein-Beltrami model of hyperbolic geometry, theconvex domain Ω bounded by a conic inherits a projectivelyinvariant hyperbolic geometry.
I The charts for the hyperbolic structure determine charts foran RP2-structure.
I Every hyperbolic manifold is convex RP2-manifold.
I A tiling of Ω = H2 in the projective model by triangles withangles π/3, π/3, π/4. The corresponding Coxeter groupcontains a finite index subgroup Γ such that Ω/Γ is a closedhyperbolic (and hence convex RP2-) surface.
![Page 39: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/39.jpg)
Hyperbolic structures as RP2-structures
I Using the Klein-Beltrami model of hyperbolic geometry, theconvex domain Ω bounded by a conic inherits a projectivelyinvariant hyperbolic geometry.
I The charts for the hyperbolic structure determine charts foran RP2-structure.
I Every hyperbolic manifold is convex RP2-manifold.
I A tiling of Ω = H2 in the projective model by triangles withangles π/3, π/3, π/4. The corresponding Coxeter groupcontains a finite index subgroup Γ such that Ω/Γ is a closedhyperbolic (and hence convex RP2-) surface.
![Page 40: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/40.jpg)
Convex RP2-structures
I χ(Σ) < 0: there will be other domains with fractal boundarydetermining convex RP2-structures M on Σ.
I (Kuiper 1954) ∂Ω is a conic and M is hyperbolic.
I (Benzecri 1960) ∂Ω is a C 1 convex curve.
I (Vinberg-Kac 1968) A triangle tiling, arising from aKac-Moody Lie algebra. The corresponding discrete group liesin SL(3,Z).
![Page 41: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/41.jpg)
Convex RP2-structures
I χ(Σ) < 0: there will be other domains with fractal boundarydetermining convex RP2-structures M on Σ.
I (Kuiper 1954) ∂Ω is a conic and M is hyperbolic.
I (Benzecri 1960) ∂Ω is a C 1 convex curve.
I (Vinberg-Kac 1968) A triangle tiling, arising from aKac-Moody Lie algebra. The corresponding discrete group liesin SL(3,Z).
![Page 42: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/42.jpg)
Convex RP2-structures
I χ(Σ) < 0: there will be other domains with fractal boundarydetermining convex RP2-structures M on Σ.
I (Kuiper 1954) ∂Ω is a conic and M is hyperbolic.
I (Benzecri 1960) ∂Ω is a C 1 convex curve.
I (Vinberg-Kac 1968) A triangle tiling, arising from aKac-Moody Lie algebra. The corresponding discrete group liesin SL(3,Z).
![Page 43: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/43.jpg)
Convex RP2-structures
I χ(Σ) < 0: there will be other domains with fractal boundarydetermining convex RP2-structures M on Σ.
I (Kuiper 1954) ∂Ω is a conic and M is hyperbolic.
I (Benzecri 1960) ∂Ω is a C 1 convex curve.
I (Vinberg-Kac 1968) A triangle tiling, arising from aKac-Moody Lie algebra. The corresponding discrete group liesin SL(3,Z).
![Page 44: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/44.jpg)
Convex RP2-structures
I χ(Σ) < 0: there will be other domains with fractal boundarydetermining convex RP2-structures M on Σ.
I (Kuiper 1954) ∂Ω is a conic and M is hyperbolic.
I (Benzecri 1960) ∂Ω is a C 1 convex curve.
I (Vinberg-Kac 1968) A triangle tiling, arising from aKac-Moody Lie algebra. The corresponding discrete group liesin SL(3,Z).
![Page 45: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/45.jpg)
Globalizing the coordinate atlas
I Coordinate changes g(C ), for C ⊂ Uα ∩ Uβ, define fibration
EΠ−→ M, fiber X , structure group G ;
I Product fibration over Uα:
Eα := Uα × XΠα−−→ Uα :
I Since C 7−→ g(C ) ∈ G is constant, the foliations of Eαdefined by projections Eα −→ X define foliation F of E ;
I Each leaf L of F is transverse to Π;
I The restriction Π|L is a covering space L −→ M.
![Page 46: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/46.jpg)
Globalizing the coordinate atlas
I Coordinate changes g(C ), for C ⊂ Uα ∩ Uβ, define fibration
EΠ−→ M, fiber X , structure group G ;
I Product fibration over Uα:
Eα := Uα × XΠα−−→ Uα :
I Since C 7−→ g(C ) ∈ G is constant, the foliations of Eαdefined by projections Eα −→ X define foliation F of E ;
I Each leaf L of F is transverse to Π;
I The restriction Π|L is a covering space L −→ M.
![Page 47: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/47.jpg)
Globalizing the coordinate atlas
I Coordinate changes g(C ), for C ⊂ Uα ∩ Uβ, define fibration
EΠ−→ M, fiber X , structure group G ;
I Product fibration over Uα:
Eα := Uα × XΠα−−→ Uα :
I Since C 7−→ g(C ) ∈ G is constant, the foliations of Eαdefined by projections Eα −→ X define foliation F of E ;
I Each leaf L of F is transverse to Π;
I The restriction Π|L is a covering space L −→ M.
![Page 48: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/48.jpg)
Globalizing the coordinate atlas
I Coordinate changes g(C ), for C ⊂ Uα ∩ Uβ, define fibration
EΠ−→ M, fiber X , structure group G ;
I Product fibration over Uα:
Eα := Uα × XΠα−−→ Uα :
I Since C 7−→ g(C ) ∈ G is constant, the foliations of Eαdefined by projections Eα −→ X define foliation F of E ;
I Each leaf L of F is transverse to Π;
I The restriction Π|L is a covering space L −→ M.
![Page 49: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/49.jpg)
Globalizing the coordinate atlas
I Coordinate changes g(C ), for C ⊂ Uα ∩ Uβ, define fibration
EΠ−→ M, fiber X , structure group G ;
I Product fibration over Uα:
Eα := Uα × XΠα−−→ Uα :
I Since C 7−→ g(C ) ∈ G is constant, the foliations of Eαdefined by projections Eα −→ X define foliation F of E ;
I Each leaf L of F is transverse to Π;
I The restriction Π|L is a covering space L −→ M.
![Page 50: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/50.jpg)
Globalizing the coordinate atlas
I Coordinate changes g(C ), for C ⊂ Uα ∩ Uβ, define fibration
EΠ−→ M, fiber X , structure group G ;
I Product fibration over Uα:
Eα := Uα × XΠα−−→ Uα :
I Since C 7−→ g(C ) ∈ G is constant, the foliations of Eαdefined by projections Eα −→ X define foliation F of E ;
I Each leaf L of F is transverse to Π;
I The restriction Π|L is a covering space L −→ M.
![Page 51: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/51.jpg)
The tangent flat (G , X )-bundle
dev
M
X
![Page 52: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/52.jpg)
The developing section
I Graph the coordinate charts Uαψα
−−→ X to obtain sections ofEα := Π−1(Uα) = Uα × X :
Uαdevα−−−→ Uα × X
I The local sections devα extend to a global section devtransverse both to Π and F.
I Such a structure is equivalent to a (G ,X )-atlas.
![Page 53: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/53.jpg)
The developing section
I Graph the coordinate charts Uαψα
−−→ X to obtain sections ofEα := Π−1(Uα) = Uα × X :
Uαdevα−−−→ Uα × X
I The local sections devα extend to a global section devtransverse both to Π and F.
I Such a structure is equivalent to a (G ,X )-atlas.
![Page 54: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/54.jpg)
The developing section
I Graph the coordinate charts Uαψα
−−→ X to obtain sections ofEα := Π−1(Uα) = Uα × X :
Uαdevα−−−→ Uα × X
I The local sections devα extend to a global section devtransverse both to Π and F.
I Such a structure is equivalent to a (G ,X )-atlas.
![Page 55: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/55.jpg)
The developing section
I Graph the coordinate charts Uαψα
−−→ X to obtain sections ofEα := Π−1(Uα) = Uα × X :
Uαdevα−−−→ Uα × X
I The local sections devα extend to a global section devtransverse both to Π and F.
I Such a structure is equivalent to a (G ,X )-atlas.
![Page 56: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/56.jpg)
The developing section of a (G , X )-structure
dev
M
X
![Page 57: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/57.jpg)
The developing section of a (G , X )-structure
dev
M
X
![Page 58: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/58.jpg)
Development, holonomy
I Let M −→ M be a universal covering with deck group π1(M).
I This structure (E ,F) is equivalent to a representation
π1(M)ρ−→ G :
I E = M × X , with π1(M)-action defined by decktransformations on M and by ρ on G .
I E = E/π1(M)
I F is the foliation defined by E −→ X .
I Sections of Π correspond to ρ-equivariant maps
Mfdev−−→ X .
I dev is F-transverse ⇐⇒ dev is a local diffeomorphism.
![Page 59: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/59.jpg)
Development, holonomy
I Let M −→ M be a universal covering with deck group π1(M).
I This structure (E ,F) is equivalent to a representation
π1(M)ρ−→ G :
I E = M × X , with π1(M)-action defined by decktransformations on M and by ρ on G .
I E = E/π1(M)
I F is the foliation defined by E −→ X .
I Sections of Π correspond to ρ-equivariant maps
Mfdev−−→ X .
I dev is F-transverse ⇐⇒ dev is a local diffeomorphism.
![Page 60: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/60.jpg)
Development, holonomy
I Let M −→ M be a universal covering with deck group π1(M).
I This structure (E ,F) is equivalent to a representation
π1(M)ρ−→ G :
I E = M × X , with π1(M)-action defined by decktransformations on M and by ρ on G .
I E = E/π1(M)
I F is the foliation defined by E −→ X .
I Sections of Π correspond to ρ-equivariant maps
Mfdev−−→ X .
I dev is F-transverse ⇐⇒ dev is a local diffeomorphism.
![Page 61: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/61.jpg)
Development, holonomy
I Let M −→ M be a universal covering with deck group π1(M).
I This structure (E ,F) is equivalent to a representation
π1(M)ρ−→ G :
I E = M × X , with π1(M)-action defined by decktransformations on M and by ρ on G .
I E = E/π1(M)
I F is the foliation defined by E −→ X .
I Sections of Π correspond to ρ-equivariant maps
Mfdev−−→ X .
I dev is F-transverse ⇐⇒ dev is a local diffeomorphism.
![Page 62: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/62.jpg)
Development, holonomy
I Let M −→ M be a universal covering with deck group π1(M).
I This structure (E ,F) is equivalent to a representation
π1(M)ρ−→ G :
I E = M × X , with π1(M)-action defined by decktransformations on M and by ρ on G .
I E = E/π1(M)
I F is the foliation defined by E −→ X .
I Sections of Π correspond to ρ-equivariant maps
Mfdev−−→ X .
I dev is F-transverse ⇐⇒ dev is a local diffeomorphism.
![Page 63: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/63.jpg)
Development, holonomy
I Let M −→ M be a universal covering with deck group π1(M).
I This structure (E ,F) is equivalent to a representation
π1(M)ρ−→ G :
I E = M × X , with π1(M)-action defined by decktransformations on M and by ρ on G .
I E = E/π1(M)
I F is the foliation defined by E −→ X .
I Sections of Π correspond to ρ-equivariant maps
Mfdev−−→ X .
I dev is F-transverse ⇐⇒ dev is a local diffeomorphism.
![Page 64: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/64.jpg)
Development, holonomy
I Let M −→ M be a universal covering with deck group π1(M).
I This structure (E ,F) is equivalent to a representation
π1(M)ρ−→ G :
I E = M × X , with π1(M)-action defined by decktransformations on M and by ρ on G .
I E = E/π1(M)
I F is the foliation defined by E −→ X .
I Sections of Π correspond to ρ-equivariant maps
Mfdev−−→ X .
I dev is F-transverse ⇐⇒ dev is a local diffeomorphism.
![Page 65: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/65.jpg)
Development, holonomy
I Let M −→ M be a universal covering with deck group π1(M).
I This structure (E ,F) is equivalent to a representation
π1(M)ρ−→ G :
I E = M × X , with π1(M)-action defined by decktransformations on M and by ρ on G .
I E = E/π1(M)
I F is the foliation defined by E −→ X .
I Sections of Π correspond to ρ-equivariant maps
Mfdev−−→ X .
I dev is F-transverse ⇐⇒ dev is a local diffeomorphism.
![Page 66: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/66.jpg)
The Ehresmann-Thurston Theorem
I Assume Σ compact. Two nearby structures with sameholonomy are isotopic: equivalent by a diffeo in Diff(M)0,
I The holonomy representation ρ of a (G ,X )-manifold M hasan open neighborhood in Hom(π1,G ) of holonomyrepresentations of nearby (G ,X )-manifolds.
![Page 67: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/67.jpg)
The Ehresmann-Thurston Theorem
I Assume Σ compact. Two nearby structures with sameholonomy are isotopic: equivalent by a diffeo in Diff(M)0,
I The holonomy representation ρ of a (G ,X )-manifold M hasan open neighborhood in Hom(π1,G ) of holonomyrepresentations of nearby (G ,X )-manifolds.
![Page 68: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/68.jpg)
The Ehresmann-Thurston Theorem
I Assume Σ compact. Two nearby structures with sameholonomy are isotopic: equivalent by a diffeo in Diff(M)0,
I The holonomy representation ρ of a (G ,X )-manifold M hasan open neighborhood in Hom(π1,G ) of holonomyrepresentations of nearby (G ,X )-manifolds.
![Page 69: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/69.jpg)
Modeling structures on representations of π1
I A marked (G ,X )-structure on Σ is a diffeomorphism Σf−→ M
where M is a (G ,X )-manifold.
I Marked (G ,X )-structures (fi ,Mi ) are isotopic ⇐⇒ ∃
isomorphism M1φ−→ M2 with φ f1 ' f2.
I Holonomy defines a local homeomorphism
D(G ,X )(Σ) :=
Marked (G ,X )-structures on Σ
/Isotopy
hol−−→ Hom(π1(Σ),G )/G
![Page 70: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/70.jpg)
Modeling structures on representations of π1
I A marked (G ,X )-structure on Σ is a diffeomorphism Σf−→ M
where M is a (G ,X )-manifold.
I Marked (G ,X )-structures (fi ,Mi ) are isotopic ⇐⇒ ∃
isomorphism M1φ−→ M2 with φ f1 ' f2.
I Holonomy defines a local homeomorphism
D(G ,X )(Σ) :=
Marked (G ,X )-structures on Σ
/Isotopy
hol−−→ Hom(π1(Σ),G )/G
![Page 71: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/71.jpg)
Modeling structures on representations of π1
I A marked (G ,X )-structure on Σ is a diffeomorphism Σf−→ M
where M is a (G ,X )-manifold.
I Marked (G ,X )-structures (fi ,Mi ) are isotopic ⇐⇒ ∃
isomorphism M1φ−→ M2 with φ f1 ' f2.
I Holonomy defines a local homeomorphism
D(G ,X )(Σ) :=
Marked (G ,X )-structures on Σ
/Isotopy
hol−−→ Hom(π1(Σ),G )/G
![Page 72: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/72.jpg)
Modeling structures on representations of π1
I A marked (G ,X )-structure on Σ is a diffeomorphism Σf−→ M
where M is a (G ,X )-manifold.
I Marked (G ,X )-structures (fi ,Mi ) are isotopic ⇐⇒ ∃
isomorphism M1φ−→ M2 with φ f1 ' f2.
I Holonomy defines a local homeomorphism
D(G ,X )(Σ) :=
Marked (G ,X )-structures on Σ
/Isotopy
hol−−→ Hom(π1(Σ),G )/G
![Page 73: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/73.jpg)
Change the marking!
I Let Σf−→ M be a marked (G ,X )-structure.
I η ∈ Diff(Σ) acts: (f ,M) 7−→ (f η,M).
I Mapping class group
Mod(Σ) := π0
(Diff(Σ)
)
acts on D(G ,X )(Σ).
I hol equivariant respecting
Mod(Σ) −→ Out(π1(Σ)
):= Aut
(π1(Σ)
)/Inn
(π1(Σ)
)
I When Σ is a closed surface (or a one-holed torus), thenπ0Diff(Σ) ∼= Out
(π1(Σ)
)(Nielsen 1918).
![Page 74: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/74.jpg)
Change the marking!
I Let Σf−→ M be a marked (G ,X )-structure.
I η ∈ Diff(Σ) acts: (f ,M) 7−→ (f η,M).
I Mapping class group
Mod(Σ) := π0
(Diff(Σ)
)
acts on D(G ,X )(Σ).
I hol equivariant respecting
Mod(Σ) −→ Out(π1(Σ)
):= Aut
(π1(Σ)
)/Inn
(π1(Σ)
)
I When Σ is a closed surface (or a one-holed torus), thenπ0Diff(Σ) ∼= Out
(π1(Σ)
)(Nielsen 1918).
![Page 75: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/75.jpg)
Change the marking!
I Let Σf−→ M be a marked (G ,X )-structure.
I η ∈ Diff(Σ) acts: (f ,M) 7−→ (f η,M).
I Mapping class group
Mod(Σ) := π0
(Diff(Σ)
)
acts on D(G ,X )(Σ).
I hol equivariant respecting
Mod(Σ) −→ Out(π1(Σ)
):= Aut
(π1(Σ)
)/Inn
(π1(Σ)
)
I When Σ is a closed surface (or a one-holed torus), thenπ0Diff(Σ) ∼= Out
(π1(Σ)
)(Nielsen 1918).
![Page 76: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/76.jpg)
Change the marking!
I Let Σf−→ M be a marked (G ,X )-structure.
I η ∈ Diff(Σ) acts: (f ,M) 7−→ (f η,M).
I Mapping class group
Mod(Σ) := π0
(Diff(Σ)
)
acts on D(G ,X )(Σ).
I hol equivariant respecting
Mod(Σ) −→ Out(π1(Σ)
):= Aut
(π1(Σ)
)/Inn
(π1(Σ)
)
I When Σ is a closed surface (or a one-holed torus), thenπ0Diff(Σ) ∼= Out
(π1(Σ)
)(Nielsen 1918).
![Page 77: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/77.jpg)
Change the marking!
I Let Σf−→ M be a marked (G ,X )-structure.
I η ∈ Diff(Σ) acts: (f ,M) 7−→ (f η,M).
I Mapping class group
Mod(Σ) := π0
(Diff(Σ)
)
acts on D(G ,X )(Σ).
I hol equivariant respecting
Mod(Σ) −→ Out(π1(Σ)
):= Aut
(π1(Σ)
)/Inn
(π1(Σ)
)
I When Σ is a closed surface (or a one-holed torus), thenπ0Diff(Σ) ∼= Out
(π1(Σ)
)(Nielsen 1918).
![Page 78: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/78.jpg)
Change the marking!
I Let Σf−→ M be a marked (G ,X )-structure.
I η ∈ Diff(Σ) acts: (f ,M) 7−→ (f η,M).
I Mapping class group
Mod(Σ) := π0
(Diff(Σ)
)
acts on D(G ,X )(Σ).
I hol equivariant respecting
Mod(Σ) −→ Out(π1(Σ)
):= Aut
(π1(Σ)
)/Inn
(π1(Σ)
)
I When Σ is a closed surface (or a one-holed torus), thenπ0Diff(Σ) ∼= Out
(π1(Σ)
)(Nielsen 1918).
![Page 79: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/79.jpg)
Marked Euclidean structures on the T 2
I Euclidean geometry: X = R2 and G = Isom(X )D(G ,X )(Σ) identifies with the upper half-plane H2:
I Point τ ∈ H2 ←→ Euclidean manifold C/〈1, τ〉.I The marking is the choice of basis 1, τ for π1(M).
I Changing the marking is the usual action of PGL(2,Z) on H2
by linear fractional transformations which is properly discrete!.
![Page 80: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/80.jpg)
Marked Euclidean structures on the T 2
I Euclidean geometry: X = R2 and G = Isom(X )D(G ,X )(Σ) identifies with the upper half-plane H2:
I Point τ ∈ H2 ←→ Euclidean manifold C/〈1, τ〉.I The marking is the choice of basis 1, τ for π1(M).
I Changing the marking is the usual action of PGL(2,Z) on H2
by linear fractional transformations which is properly discrete!.
![Page 81: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/81.jpg)
Marked Euclidean structures on the T 2
I Euclidean geometry: X = R2 and G = Isom(X )D(G ,X )(Σ) identifies with the upper half-plane H2:
I Point τ ∈ H2 ←→ Euclidean manifold C/〈1, τ〉.I The marking is the choice of basis 1, τ for π1(M).
I Changing the marking is the usual action of PGL(2,Z) on H2
by linear fractional transformations which is properly discrete!.
![Page 82: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/82.jpg)
Marked Euclidean structures on the T 2
I Euclidean geometry: X = R2 and G = Isom(X )D(G ,X )(Σ) identifies with the upper half-plane H2:
I Point τ ∈ H2 ←→ Euclidean manifold C/〈1, τ〉.I The marking is the choice of basis 1, τ for π1(M).
I Changing the marking is the usual action of PGL(2,Z) on H2
by linear fractional transformations which is properly discrete!.
![Page 83: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/83.jpg)
Marked Euclidean structures on the T 2
I Euclidean geometry: X = R2 and G = Isom(X )D(G ,X )(Σ) identifies with the upper half-plane H2:
I Point τ ∈ H2 ←→ Euclidean manifold C/〈1, τ〉.I The marking is the choice of basis 1, τ for π1(M).
I Changing the marking is the usual action of PGL(2,Z) on H2
by linear fractional transformations which is properly discrete!.
![Page 84: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/84.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 85: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/85.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 86: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/86.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 87: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/87.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 88: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/88.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 89: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/89.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 90: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/90.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 91: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/91.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 92: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/92.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 93: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/93.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 94: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/94.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 95: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/95.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 96: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/96.jpg)
Complete affine structures on the 2-torus
I Kuiper (1954) Every complete affine closed orientable2-manifold is equivalent to either:
I Euclidean: R2/Λ, where Λ is a lattice of translations(all are affinely equivalent);
I Polynomial deformation R2/(f Λ f −1), where
(x , y)f−→ (x + y 2, y).
![Page 97: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/97.jpg)
Chaotic dynamics on the deformation space
I Usually Mod(Σ) too dynamicly interesting to form a quotient.
I (Baues 2000) Deformation space homeomorphic R2, whereorigin (0, 0) corresponds to Euclidean structure;
I Mapping class group action is linear GL(2,Z)-action on R2 —chaotic!
I The orbit space — the moduli space of complete affinecompact orientable 2-manifolds is non-Hausdorff andintractable. (Even though the corresponding representationsare discrete embeddings.)
I In contrast, Mod(Σ) can act properly discrete even fornon-discrete representations: hyperbolic structures on T 2 withsingle cone point.
![Page 98: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/98.jpg)
Chaotic dynamics on the deformation space
I Usually Mod(Σ) too dynamicly interesting to form a quotient.
I (Baues 2000) Deformation space homeomorphic R2, whereorigin (0, 0) corresponds to Euclidean structure;
I Mapping class group action is linear GL(2,Z)-action on R2 —chaotic!
I The orbit space — the moduli space of complete affinecompact orientable 2-manifolds is non-Hausdorff andintractable. (Even though the corresponding representationsare discrete embeddings.)
I In contrast, Mod(Σ) can act properly discrete even fornon-discrete representations: hyperbolic structures on T 2 withsingle cone point.
![Page 99: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/99.jpg)
Chaotic dynamics on the deformation space
I Usually Mod(Σ) too dynamicly interesting to form a quotient.
I (Baues 2000) Deformation space homeomorphic R2, whereorigin (0, 0) corresponds to Euclidean structure;
I Mapping class group action is linear GL(2,Z)-action on R2 —chaotic!
I The orbit space — the moduli space of complete affinecompact orientable 2-manifolds is non-Hausdorff andintractable. (Even though the corresponding representationsare discrete embeddings.)
I In contrast, Mod(Σ) can act properly discrete even fornon-discrete representations: hyperbolic structures on T 2 withsingle cone point.
![Page 100: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/100.jpg)
Chaotic dynamics on the deformation space
I Usually Mod(Σ) too dynamicly interesting to form a quotient.
I (Baues 2000) Deformation space homeomorphic R2, whereorigin (0, 0) corresponds to Euclidean structure;
I Mapping class group action is linear GL(2,Z)-action on R2 —chaotic!
I The orbit space — the moduli space of complete affinecompact orientable 2-manifolds is non-Hausdorff andintractable. (Even though the corresponding representationsare discrete embeddings.)
I In contrast, Mod(Σ) can act properly discrete even fornon-discrete representations: hyperbolic structures on T 2 withsingle cone point.
![Page 101: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/101.jpg)
Chaotic dynamics on the deformation space
I Usually Mod(Σ) too dynamicly interesting to form a quotient.
I (Baues 2000) Deformation space homeomorphic R2, whereorigin (0, 0) corresponds to Euclidean structure;
I Mapping class group action is linear GL(2,Z)-action on R2 —chaotic!
I The orbit space — the moduli space of complete affinecompact orientable 2-manifolds is non-Hausdorff andintractable. (Even though the corresponding representationsare discrete embeddings.)
I In contrast, Mod(Σ) can act properly discrete even fornon-discrete representations: hyperbolic structures on T 2 withsingle cone point.
![Page 102: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/102.jpg)
Chaotic dynamics on the deformation space
I Usually Mod(Σ) too dynamicly interesting to form a quotient.
I (Baues 2000) Deformation space homeomorphic R2, whereorigin (0, 0) corresponds to Euclidean structure;
I Mapping class group action is linear GL(2,Z)-action on R2 —chaotic!
I The orbit space — the moduli space of complete affinecompact orientable 2-manifolds is non-Hausdorff andintractable. (Even though the corresponding representationsare discrete embeddings.)
I In contrast, Mod(Σ) can act properly discrete even fornon-discrete representations: hyperbolic structures on T 2 withsingle cone point.
![Page 103: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/103.jpg)
Fricke spaces of hyperbolic structures
I Hyperbolic geometry: X = H2 and G = Isom(X )D(G ,X )(Σ) is Fricke space F(Σ) of isotopy classes of markedhyperbolic structures Σ −→ M.
I
F(Σ)hol→ Hom(π,G ))/G
embeds F(Σ) as a connected component.
I Image comprises discrete embeddings πρ→ G .
I Equivalently, ρ(γ) is hyperbolic if γ 6= 1.
I (Fricke-Klein ?) F(Σ) diffeomorphic to R−3χ(Σ).
I Uniformization: F(Σ) corresponds to marked conformal
structures: Teichmuller space.
I Mod(Σ) acts properly on F(Σ) with quotient Riemann’smoduli space of Riemann surfaces of fixed topology Σ.
![Page 104: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/104.jpg)
Fricke spaces of hyperbolic structures
I Hyperbolic geometry: X = H2 and G = Isom(X )D(G ,X )(Σ) is Fricke space F(Σ) of isotopy classes of markedhyperbolic structures Σ −→ M.
I
F(Σ)hol→ Hom(π,G ))/G
embeds F(Σ) as a connected component.
I Image comprises discrete embeddings πρ→ G .
I Equivalently, ρ(γ) is hyperbolic if γ 6= 1.
I (Fricke-Klein ?) F(Σ) diffeomorphic to R−3χ(Σ).
I Uniformization: F(Σ) corresponds to marked conformal
structures: Teichmuller space.
I Mod(Σ) acts properly on F(Σ) with quotient Riemann’smoduli space of Riemann surfaces of fixed topology Σ.
![Page 105: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/105.jpg)
Fricke spaces of hyperbolic structures
I Hyperbolic geometry: X = H2 and G = Isom(X )D(G ,X )(Σ) is Fricke space F(Σ) of isotopy classes of markedhyperbolic structures Σ −→ M.
I
F(Σ)hol→ Hom(π,G ))/G
embeds F(Σ) as a connected component.
I Image comprises discrete embeddings πρ→ G .
I Equivalently, ρ(γ) is hyperbolic if γ 6= 1.
I (Fricke-Klein ?) F(Σ) diffeomorphic to R−3χ(Σ).
I Uniformization: F(Σ) corresponds to marked conformal
structures: Teichmuller space.
I Mod(Σ) acts properly on F(Σ) with quotient Riemann’smoduli space of Riemann surfaces of fixed topology Σ.
![Page 106: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/106.jpg)
Fricke spaces of hyperbolic structures
I Hyperbolic geometry: X = H2 and G = Isom(X )D(G ,X )(Σ) is Fricke space F(Σ) of isotopy classes of markedhyperbolic structures Σ −→ M.
I
F(Σ)hol→ Hom(π,G ))/G
embeds F(Σ) as a connected component.
I Image comprises discrete embeddings πρ→ G .
I Equivalently, ρ(γ) is hyperbolic if γ 6= 1.
I (Fricke-Klein ?) F(Σ) diffeomorphic to R−3χ(Σ).
I Uniformization: F(Σ) corresponds to marked conformal
structures: Teichmuller space.
I Mod(Σ) acts properly on F(Σ) with quotient Riemann’smoduli space of Riemann surfaces of fixed topology Σ.
![Page 107: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/107.jpg)
Fricke spaces of hyperbolic structures
I Hyperbolic geometry: X = H2 and G = Isom(X )D(G ,X )(Σ) is Fricke space F(Σ) of isotopy classes of markedhyperbolic structures Σ −→ M.
I
F(Σ)hol→ Hom(π,G ))/G
embeds F(Σ) as a connected component.
I Image comprises discrete embeddings πρ→ G .
I Equivalently, ρ(γ) is hyperbolic if γ 6= 1.
I (Fricke-Klein ?) F(Σ) diffeomorphic to R−3χ(Σ).
I Uniformization: F(Σ) corresponds to marked conformal
structures: Teichmuller space.
I Mod(Σ) acts properly on F(Σ) with quotient Riemann’smoduli space of Riemann surfaces of fixed topology Σ.
![Page 108: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/108.jpg)
Fricke spaces of hyperbolic structures
I Hyperbolic geometry: X = H2 and G = Isom(X )D(G ,X )(Σ) is Fricke space F(Σ) of isotopy classes of markedhyperbolic structures Σ −→ M.
I
F(Σ)hol→ Hom(π,G ))/G
embeds F(Σ) as a connected component.
I Image comprises discrete embeddings πρ→ G .
I Equivalently, ρ(γ) is hyperbolic if γ 6= 1.
I (Fricke-Klein ?) F(Σ) diffeomorphic to R−3χ(Σ).
I Uniformization: F(Σ) corresponds to marked conformal
structures: Teichmuller space.
I Mod(Σ) acts properly on F(Σ) with quotient Riemann’smoduli space of Riemann surfaces of fixed topology Σ.
![Page 109: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/109.jpg)
Fricke spaces of hyperbolic structures
I Hyperbolic geometry: X = H2 and G = Isom(X )D(G ,X )(Σ) is Fricke space F(Σ) of isotopy classes of markedhyperbolic structures Σ −→ M.
I
F(Σ)hol→ Hom(π,G ))/G
embeds F(Σ) as a connected component.
I Image comprises discrete embeddings πρ→ G .
I Equivalently, ρ(γ) is hyperbolic if γ 6= 1.
I (Fricke-Klein ?) F(Σ) diffeomorphic to R−3χ(Σ).
I Uniformization: F(Σ) corresponds to marked conformal
structures: Teichmuller space.
I Mod(Σ) acts properly on F(Σ) with quotient Riemann’smoduli space of Riemann surfaces of fixed topology Σ.
![Page 110: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/110.jpg)
Fricke spaces of hyperbolic structures
I Hyperbolic geometry: X = H2 and G = Isom(X )D(G ,X )(Σ) is Fricke space F(Σ) of isotopy classes of markedhyperbolic structures Σ −→ M.
I
F(Σ)hol→ Hom(π,G ))/G
embeds F(Σ) as a connected component.
I Image comprises discrete embeddings πρ→ G .
I Equivalently, ρ(γ) is hyperbolic if γ 6= 1.
I (Fricke-Klein ?) F(Σ) diffeomorphic to R−3χ(Σ).
I Uniformization: F(Σ) corresponds to marked conformal
structures: Teichmuller space.
I Mod(Σ) acts properly on F(Σ) with quotient Riemann’smoduli space of Riemann surfaces of fixed topology Σ.
![Page 111: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/111.jpg)
Classification of RP2-surfaces
I (Goldman 1990) Isotopy classes of marked convexRP2-structures on Σ form deformation space
C(Σ) ≈ R−8χ(Σ).
I Mod(Σ) acts properly on C(Σ).
I (Choi-Goldman 1993, conjectured by Hitchin 1992)
C(Σ)hol→ Hom(π,SL(3,R))/SL(3,R).
embedding onto connected component.
I (Choi 1986) M decomposes canonically into convex pieceswith geodesic boundary.
I Complete and explicit description:
D(PGL(3,R),RP2)(Σ) ≈ C(Σ)× Z ≈ R−8χ(Σ) × Z
![Page 112: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/112.jpg)
Classification of RP2-surfaces
I (Goldman 1990) Isotopy classes of marked convexRP2-structures on Σ form deformation space
C(Σ) ≈ R−8χ(Σ).
I Mod(Σ) acts properly on C(Σ).
I (Choi-Goldman 1993, conjectured by Hitchin 1992)
C(Σ)hol→ Hom(π,SL(3,R))/SL(3,R).
embedding onto connected component.
I (Choi 1986) M decomposes canonically into convex pieceswith geodesic boundary.
I Complete and explicit description:
D(PGL(3,R),RP2)(Σ) ≈ C(Σ)× Z ≈ R−8χ(Σ) × Z
![Page 113: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/113.jpg)
Classification of RP2-surfaces
I (Goldman 1990) Isotopy classes of marked convexRP2-structures on Σ form deformation space
C(Σ) ≈ R−8χ(Σ).
I Mod(Σ) acts properly on C(Σ).
I (Choi-Goldman 1993, conjectured by Hitchin 1992)
C(Σ)hol→ Hom(π,SL(3,R))/SL(3,R).
embedding onto connected component.
I (Choi 1986) M decomposes canonically into convex pieceswith geodesic boundary.
I Complete and explicit description:
D(PGL(3,R),RP2)(Σ) ≈ C(Σ)× Z ≈ R−8χ(Σ) × Z
![Page 114: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/114.jpg)
Classification of RP2-surfaces
I (Goldman 1990) Isotopy classes of marked convexRP2-structures on Σ form deformation space
C(Σ) ≈ R−8χ(Σ).
I Mod(Σ) acts properly on C(Σ).
I (Choi-Goldman 1993, conjectured by Hitchin 1992)
C(Σ)hol→ Hom(π,SL(3,R))/SL(3,R).
embedding onto connected component.
I (Choi 1986) M decomposes canonically into convex pieceswith geodesic boundary.
I Complete and explicit description:
D(PGL(3,R),RP2)(Σ) ≈ C(Σ)× Z ≈ R−8χ(Σ) × Z
![Page 115: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/115.jpg)
Classification of RP2-surfaces
I (Goldman 1990) Isotopy classes of marked convexRP2-structures on Σ form deformation space
C(Σ) ≈ R−8χ(Σ).
I Mod(Σ) acts properly on C(Σ).
I (Choi-Goldman 1993, conjectured by Hitchin 1992)
C(Σ)hol→ Hom(π,SL(3,R))/SL(3,R).
embedding onto connected component.
I (Choi 1986) M decomposes canonically into convex pieceswith geodesic boundary.
I Complete and explicit description:
D(PGL(3,R),RP2)(Σ) ≈ C(Σ)× Z ≈ R−8χ(Σ) × Z
![Page 116: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/116.jpg)
Classification of RP2-surfaces
I (Goldman 1990) Isotopy classes of marked convexRP2-structures on Σ form deformation space
C(Σ) ≈ R−8χ(Σ).
I Mod(Σ) acts properly on C(Σ).
I (Choi-Goldman 1993, conjectured by Hitchin 1992)
C(Σ)hol→ Hom(π,SL(3,R))/SL(3,R).
embedding onto connected component.
I (Choi 1986) M decomposes canonically into convex pieceswith geodesic boundary.
I Complete and explicit description:
D(PGL(3,R),RP2)(Σ) ≈ C(Σ)× Z ≈ R−8χ(Σ) × Z
![Page 117: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/117.jpg)
Recent developments
I (Hitchin 1990) G split R-semisimple Lie group:Hom(π,G )/G always contains connected component
HG (Σ) ≈ R− dim(G)χ(Σ).
I (Labourie 2003) Every Hitchin representation is aquasi-isometric discrete embedding π −→ G .
I Mod(Σ) acts properly on HG (Σ).
I ∀γ 6= 1, ρ(γ) is positive hyperbolic.
I (Labourie, Guichard, Fock-Goncharov) Hitchin representationscharacterized by positivity condition. Limit set is a Holdercontinuous closed curve in RPn.
I (Benoist 2001) In all dimensions, geodesic flow of Hilbertmetric of strictly convex projective structures is Anosov.
![Page 118: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/118.jpg)
Recent developments
I (Hitchin 1990) G split R-semisimple Lie group:Hom(π,G )/G always contains connected component
HG (Σ) ≈ R− dim(G)χ(Σ).
I (Labourie 2003) Every Hitchin representation is aquasi-isometric discrete embedding π −→ G .
I Mod(Σ) acts properly on HG (Σ).
I ∀γ 6= 1, ρ(γ) is positive hyperbolic.
I (Labourie, Guichard, Fock-Goncharov) Hitchin representationscharacterized by positivity condition. Limit set is a Holdercontinuous closed curve in RPn.
I (Benoist 2001) In all dimensions, geodesic flow of Hilbertmetric of strictly convex projective structures is Anosov.
![Page 119: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/119.jpg)
Recent developments
I (Hitchin 1990) G split R-semisimple Lie group:Hom(π,G )/G always contains connected component
HG (Σ) ≈ R− dim(G)χ(Σ).
I (Labourie 2003) Every Hitchin representation is aquasi-isometric discrete embedding π −→ G .
I Mod(Σ) acts properly on HG (Σ).
I ∀γ 6= 1, ρ(γ) is positive hyperbolic.
I (Labourie, Guichard, Fock-Goncharov) Hitchin representationscharacterized by positivity condition. Limit set is a Holdercontinuous closed curve in RPn.
I (Benoist 2001) In all dimensions, geodesic flow of Hilbertmetric of strictly convex projective structures is Anosov.
![Page 120: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/120.jpg)
Recent developments
I (Hitchin 1990) G split R-semisimple Lie group:Hom(π,G )/G always contains connected component
HG (Σ) ≈ R− dim(G)χ(Σ).
I (Labourie 2003) Every Hitchin representation is aquasi-isometric discrete embedding π −→ G .
I Mod(Σ) acts properly on HG (Σ).
I ∀γ 6= 1, ρ(γ) is positive hyperbolic.
I (Labourie, Guichard, Fock-Goncharov) Hitchin representationscharacterized by positivity condition. Limit set is a Holdercontinuous closed curve in RPn.
I (Benoist 2001) In all dimensions, geodesic flow of Hilbertmetric of strictly convex projective structures is Anosov.
![Page 121: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/121.jpg)
Recent developments
I (Hitchin 1990) G split R-semisimple Lie group:Hom(π,G )/G always contains connected component
HG (Σ) ≈ R− dim(G)χ(Σ).
I (Labourie 2003) Every Hitchin representation is aquasi-isometric discrete embedding π −→ G .
I Mod(Σ) acts properly on HG (Σ).
I ∀γ 6= 1, ρ(γ) is positive hyperbolic.
I (Labourie, Guichard, Fock-Goncharov) Hitchin representationscharacterized by positivity condition. Limit set is a Holdercontinuous closed curve in RPn.
I (Benoist 2001) In all dimensions, geodesic flow of Hilbertmetric of strictly convex projective structures is Anosov.
![Page 122: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/122.jpg)
Recent developments
I (Hitchin 1990) G split R-semisimple Lie group:Hom(π,G )/G always contains connected component
HG (Σ) ≈ R− dim(G)χ(Σ).
I (Labourie 2003) Every Hitchin representation is aquasi-isometric discrete embedding π −→ G .
I Mod(Σ) acts properly on HG (Σ).
I ∀γ 6= 1, ρ(γ) is positive hyperbolic.
I (Labourie, Guichard, Fock-Goncharov) Hitchin representationscharacterized by positivity condition. Limit set is a Holdercontinuous closed curve in RPn.
I (Benoist 2001) In all dimensions, geodesic flow of Hilbertmetric of strictly convex projective structures is Anosov.
![Page 123: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/123.jpg)
Recent developments
I (Hitchin 1990) G split R-semisimple Lie group:Hom(π,G )/G always contains connected component
HG (Σ) ≈ R− dim(G)χ(Σ).
I (Labourie 2003) Every Hitchin representation is aquasi-isometric discrete embedding π −→ G .
I Mod(Σ) acts properly on HG (Σ).
I ∀γ 6= 1, ρ(γ) is positive hyperbolic.
I (Labourie, Guichard, Fock-Goncharov) Hitchin representationscharacterized by positivity condition. Limit set is a Holdercontinuous closed curve in RPn.
I (Benoist 2001) In all dimensions, geodesic flow of Hilbertmetric of strictly convex projective structures is Anosov.
![Page 124: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/124.jpg)
Representations and their symmetriesI Let π = 〈X1, . . . ,Xn〉 be finitely generated and G Lie group:.I The set Hom(π,G ) of homomorphisms
π −→ G
admits an action of Aut(π)× Aut(G ):
πφ−→ π
ρ−→ G
α−→ G
where (φ, α) ∈ Aut(π)× Aut(G ), ρ ∈ Hom(π,G ).I The quotient
Hom(π,G )/G := Hom(π,G )/(1 × Inn(G )
)
under the subgroup
1 × Inn(G ) ⊂ Aut(π)× Aut(G )
admits an action of
Out(π) := Aut(π)/Inn(π).
![Page 125: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/125.jpg)
Representations and their symmetriesI Let π = 〈X1, . . . ,Xn〉 be finitely generated and G Lie group:.I The set Hom(π,G ) of homomorphisms
π −→ G
admits an action of Aut(π)× Aut(G ):
πφ−→ π
ρ−→ G
α−→ G
where (φ, α) ∈ Aut(π)× Aut(G ), ρ ∈ Hom(π,G ).I The quotient
Hom(π,G )/G := Hom(π,G )/(1 × Inn(G )
)
under the subgroup
1 × Inn(G ) ⊂ Aut(π)× Aut(G )
admits an action of
Out(π) := Aut(π)/Inn(π).
![Page 126: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/126.jpg)
Representations and their symmetriesI Let π = 〈X1, . . . ,Xn〉 be finitely generated and G Lie group:.I The set Hom(π,G ) of homomorphisms
π −→ G
admits an action of Aut(π)× Aut(G ):
πφ−→ π
ρ−→ G
α−→ G
where (φ, α) ∈ Aut(π)× Aut(G ), ρ ∈ Hom(π,G ).I The quotient
Hom(π,G )/G := Hom(π,G )/(1 × Inn(G )
)
under the subgroup
1 × Inn(G ) ⊂ Aut(π)× Aut(G )
admits an action of
Out(π) := Aut(π)/Inn(π).
![Page 127: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/127.jpg)
Representations and their symmetriesI Let π = 〈X1, . . . ,Xn〉 be finitely generated and G Lie group:.I The set Hom(π,G ) of homomorphisms
π −→ G
admits an action of Aut(π)× Aut(G ):
πφ−→ π
ρ−→ G
α−→ G
where (φ, α) ∈ Aut(π)× Aut(G ), ρ ∈ Hom(π,G ).I The quotient
Hom(π,G )/G := Hom(π,G )/(1 × Inn(G )
)
under the subgroup
1 × Inn(G ) ⊂ Aut(π)× Aut(G )
admits an action of
Out(π) := Aut(π)/Inn(π).
![Page 128: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/128.jpg)
Algebraic structure of representation spaces
I G : algebraic Lie group.
I ρ 7−→(ρ(X1) . . . ρ(Xn)
)embeds Hom(π,G ) onto an algebraic
subset of G n.
I Algebraic structure is X1, . . . ,Xn-independent andAut(π)× Aut(G )-invariant.
I Geometric Invariant Theory quotient Hom(π,G )//G isOut(π)-invariant.
I Coordinate ring is the invariant subring
C[Hom(π,G )//G ] = C[Hom(π,G )]G ⊂ C[Hom(π,G )].
I Examples are functions fα, associated to:I A conjugacy class [α], where α ∈ π;
I An Inn(G )-invariant function Gf−→ R.
![Page 129: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/129.jpg)
Algebraic structure of representation spaces
I G : algebraic Lie group.
I ρ 7−→(ρ(X1) . . . ρ(Xn)
)embeds Hom(π,G ) onto an algebraic
subset of G n.
I Algebraic structure is X1, . . . ,Xn-independent andAut(π)× Aut(G )-invariant.
I Geometric Invariant Theory quotient Hom(π,G )//G isOut(π)-invariant.
I Coordinate ring is the invariant subring
C[Hom(π,G )//G ] = C[Hom(π,G )]G ⊂ C[Hom(π,G )].
I Examples are functions fα, associated to:I A conjugacy class [α], where α ∈ π;
I An Inn(G )-invariant function Gf−→ R.
![Page 130: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/130.jpg)
Algebraic structure of representation spaces
I G : algebraic Lie group.
I ρ 7−→(ρ(X1) . . . ρ(Xn)
)embeds Hom(π,G ) onto an algebraic
subset of G n.
I Algebraic structure is X1, . . . ,Xn-independent andAut(π)× Aut(G )-invariant.
I Geometric Invariant Theory quotient Hom(π,G )//G isOut(π)-invariant.
I Coordinate ring is the invariant subring
C[Hom(π,G )//G ] = C[Hom(π,G )]G ⊂ C[Hom(π,G )].
I Examples are functions fα, associated to:I A conjugacy class [α], where α ∈ π;
I An Inn(G )-invariant function Gf−→ R.
![Page 131: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/131.jpg)
Algebraic structure of representation spaces
I G : algebraic Lie group.
I ρ 7−→(ρ(X1) . . . ρ(Xn)
)embeds Hom(π,G ) onto an algebraic
subset of G n.
I Algebraic structure is X1, . . . ,Xn-independent andAut(π)× Aut(G )-invariant.
I Geometric Invariant Theory quotient Hom(π,G )//G isOut(π)-invariant.
I Coordinate ring is the invariant subring
C[Hom(π,G )//G ] = C[Hom(π,G )]G ⊂ C[Hom(π,G )].
I Examples are functions fα, associated to:I A conjugacy class [α], where α ∈ π;
I An Inn(G )-invariant function Gf−→ R.
![Page 132: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/132.jpg)
Algebraic structure of representation spaces
I G : algebraic Lie group.
I ρ 7−→(ρ(X1) . . . ρ(Xn)
)embeds Hom(π,G ) onto an algebraic
subset of G n.
I Algebraic structure is X1, . . . ,Xn-independent andAut(π)× Aut(G )-invariant.
I Geometric Invariant Theory quotient Hom(π,G )//G isOut(π)-invariant.
I Coordinate ring is the invariant subring
C[Hom(π,G )//G ] = C[Hom(π,G )]G ⊂ C[Hom(π,G )].
I Examples are functions fα, associated to:I A conjugacy class [α], where α ∈ π;
I An Inn(G )-invariant function Gf−→ R.
![Page 133: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/133.jpg)
Algebraic structure of representation spaces
I G : algebraic Lie group.
I ρ 7−→(ρ(X1) . . . ρ(Xn)
)embeds Hom(π,G ) onto an algebraic
subset of G n.
I Algebraic structure is X1, . . . ,Xn-independent andAut(π)× Aut(G )-invariant.
I Geometric Invariant Theory quotient Hom(π,G )//G isOut(π)-invariant.
I Coordinate ring is the invariant subring
C[Hom(π,G )//G ] = C[Hom(π,G )]G ⊂ C[Hom(π,G )].
I Examples are functions fα, associated to:I A conjugacy class [α], where α ∈ π;
I An Inn(G )-invariant function Gf−→ R.
![Page 134: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/134.jpg)
Algebraic structure of representation spaces
I G : algebraic Lie group.
I ρ 7−→(ρ(X1) . . . ρ(Xn)
)embeds Hom(π,G ) onto an algebraic
subset of G n.
I Algebraic structure is X1, . . . ,Xn-independent andAut(π)× Aut(G )-invariant.
I Geometric Invariant Theory quotient Hom(π,G )//G isOut(π)-invariant.
I Coordinate ring is the invariant subring
C[Hom(π,G )//G ] = C[Hom(π,G )]G ⊂ C[Hom(π,G )].
I Examples are functions fα, associated to:I A conjugacy class [α], where α ∈ π;
I An Inn(G )-invariant function Gf−→ R.
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Character functions fα on representation varieties
I Invariant function Gf−→ R =⇒ Function fα on Hom(π,G )/G
Hom(π,G )/Gfα−→ R
[ρ] 7−→ f(ρ(α)
)
Conjugacy class of α ∈ π corresponds to free homotopy classof closed oriented loop α ⊂ Σ.
I These functions generate the coordinate ring.
I Example: Trace GL(n,R)tr−→ R
I Another example: Displacement length on SL(2,R):
`(A) := minx∈H2
d(x ,A(x)
)
I If A is hyperbolic, tr(A) = ±2 cosh(`(A)/2
).
![Page 136: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/136.jpg)
Character functions fα on representation varieties
I Invariant function Gf−→ R =⇒ Function fα on Hom(π,G )/G
Hom(π,G )/Gfα−→ R
[ρ] 7−→ f(ρ(α)
)
Conjugacy class of α ∈ π corresponds to free homotopy classof closed oriented loop α ⊂ Σ.
I These functions generate the coordinate ring.
I Example: Trace GL(n,R)tr−→ R
I Another example: Displacement length on SL(2,R):
`(A) := minx∈H2
d(x ,A(x)
)
I If A is hyperbolic, tr(A) = ±2 cosh(`(A)/2
).
![Page 137: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/137.jpg)
Character functions fα on representation varieties
I Invariant function Gf−→ R =⇒ Function fα on Hom(π,G )/G
Hom(π,G )/Gfα−→ R
[ρ] 7−→ f(ρ(α)
)
Conjugacy class of α ∈ π corresponds to free homotopy classof closed oriented loop α ⊂ Σ.
I These functions generate the coordinate ring.
I Example: Trace GL(n,R)tr−→ R
I Another example: Displacement length on SL(2,R):
`(A) := minx∈H2
d(x ,A(x)
)
I If A is hyperbolic, tr(A) = ±2 cosh(`(A)/2
).
![Page 138: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/138.jpg)
Character functions fα on representation varieties
I Invariant function Gf−→ R =⇒ Function fα on Hom(π,G )/G
Hom(π,G )/Gfα−→ R
[ρ] 7−→ f(ρ(α)
)
Conjugacy class of α ∈ π corresponds to free homotopy classof closed oriented loop α ⊂ Σ.
I These functions generate the coordinate ring.
I Example: Trace GL(n,R)tr−→ R
I Another example: Displacement length on SL(2,R):
`(A) := minx∈H2
d(x ,A(x)
)
I If A is hyperbolic, tr(A) = ±2 cosh(`(A)/2
).
![Page 139: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/139.jpg)
Character functions fα on representation varieties
I Invariant function Gf−→ R =⇒ Function fα on Hom(π,G )/G
Hom(π,G )/Gfα−→ R
[ρ] 7−→ f(ρ(α)
)
Conjugacy class of α ∈ π corresponds to free homotopy classof closed oriented loop α ⊂ Σ.
I These functions generate the coordinate ring.
I Example: Trace GL(n,R)tr−→ R
I Another example: Displacement length on SL(2,R):
`(A) := minx∈H2
d(x ,A(x)
)
I If A is hyperbolic, tr(A) = ±2 cosh(`(A)/2
).
![Page 140: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/140.jpg)
Character functions fα on representation varieties
I Invariant function Gf−→ R =⇒ Function fα on Hom(π,G )/G
Hom(π,G )/Gfα−→ R
[ρ] 7−→ f(ρ(α)
)
Conjugacy class of α ∈ π corresponds to free homotopy classof closed oriented loop α ⊂ Σ.
I These functions generate the coordinate ring.
I Example: Trace GL(n,R)tr−→ R
I Another example: Displacement length on SL(2,R):
`(A) := minx∈H2
d(x ,A(x)
)
I If A is hyperbolic, tr(A) = ±2 cosh(`(A)/2
).
![Page 141: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/141.jpg)
Invariant functions in PGL(3, R) ∼= SL(3, R)
I Restrict to the subset Hyp+ ⊂ SL(3,R) consisting of positive
hyperbolic elements (diagonalizable over R):
A ∼
λ1 0 00 λ2 00 0 λ3
where λ1 > λ2 > λ3 > 0 and λ1λ2λ3 = 1.
I The Hilbert displacement corresponds to the invariantfunction
`(A) := log(λ1/λ3) = log(λ1)− log(λ3)
I The bulging parameter is the invariant function
β(A) := log(λ2)
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Invariant functions in PGL(3, R) ∼= SL(3, R)
I Restrict to the subset Hyp+ ⊂ SL(3,R) consisting of positive
hyperbolic elements (diagonalizable over R):
A ∼
λ1 0 00 λ2 00 0 λ3
where λ1 > λ2 > λ3 > 0 and λ1λ2λ3 = 1.
I The Hilbert displacement corresponds to the invariantfunction
`(A) := log(λ1/λ3) = log(λ1)− log(λ3)
I The bulging parameter is the invariant function
β(A) := log(λ2)
![Page 143: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/143.jpg)
Invariant functions in PGL(3, R) ∼= SL(3, R)
I Restrict to the subset Hyp+ ⊂ SL(3,R) consisting of positive
hyperbolic elements (diagonalizable over R):
A ∼
λ1 0 00 λ2 00 0 λ3
where λ1 > λ2 > λ3 > 0 and λ1λ2λ3 = 1.
I The Hilbert displacement corresponds to the invariantfunction
`(A) := log(λ1/λ3) = log(λ1)− log(λ3)
I The bulging parameter is the invariant function
β(A) := log(λ2)
![Page 144: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/144.jpg)
Invariant functions in PGL(3, R) ∼= SL(3, R)
I Restrict to the subset Hyp+ ⊂ SL(3,R) consisting of positive
hyperbolic elements (diagonalizable over R):
A ∼
λ1 0 00 λ2 00 0 λ3
where λ1 > λ2 > λ3 > 0 and λ1λ2λ3 = 1.
I The Hilbert displacement corresponds to the invariantfunction
`(A) := log(λ1/λ3) = log(λ1)− log(λ3)
I The bulging parameter is the invariant function
β(A) := log(λ2)
![Page 145: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/145.jpg)
Marked length spectra in F(Σ) and C(Σ)
I On F(Σ), `α associates to a marked hyperbolic surface Σ ≈ M
length of the unique closed geodesic homotopic to α in M.
I On C(Σ), `α associates to a marked convex RP2-surfaceΣ ≈ M the Hilbert length of the unique closed geodesichomotopic to α in M.
I (Fricke-Klein ?) The marked length spectrum characterizeshyperbolic structures in F(Σ).
I (Inkang Kim, 2001) The marked Hilbert length spectrum
characterizes RP2-structures in C(Σ).
![Page 146: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/146.jpg)
Marked length spectra in F(Σ) and C(Σ)
I On F(Σ), `α associates to a marked hyperbolic surface Σ ≈ M
length of the unique closed geodesic homotopic to α in M.
I On C(Σ), `α associates to a marked convex RP2-surfaceΣ ≈ M the Hilbert length of the unique closed geodesichomotopic to α in M.
I (Fricke-Klein ?) The marked length spectrum characterizeshyperbolic structures in F(Σ).
I (Inkang Kim, 2001) The marked Hilbert length spectrum
characterizes RP2-structures in C(Σ).
![Page 147: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/147.jpg)
Marked length spectra in F(Σ) and C(Σ)
I On F(Σ), `α associates to a marked hyperbolic surface Σ ≈ M
length of the unique closed geodesic homotopic to α in M.
I On C(Σ), `α associates to a marked convex RP2-surfaceΣ ≈ M the Hilbert length of the unique closed geodesichomotopic to α in M.
I (Fricke-Klein ?) The marked length spectrum characterizeshyperbolic structures in F(Σ).
I (Inkang Kim, 2001) The marked Hilbert length spectrum
characterizes RP2-structures in C(Σ).
![Page 148: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/148.jpg)
Marked length spectra in F(Σ) and C(Σ)
I On F(Σ), `α associates to a marked hyperbolic surface Σ ≈ M
length of the unique closed geodesic homotopic to α in M.
I On C(Σ), `α associates to a marked convex RP2-surfaceΣ ≈ M the Hilbert length of the unique closed geodesichomotopic to α in M.
I (Fricke-Klein ?) The marked length spectrum characterizeshyperbolic structures in F(Σ).
I (Inkang Kim, 2001) The marked Hilbert length spectrum
characterizes RP2-structures in C(Σ).
![Page 149: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/149.jpg)
Marked length spectra in F(Σ) and C(Σ)
I On F(Σ), `α associates to a marked hyperbolic surface Σ ≈ M
length of the unique closed geodesic homotopic to α in M.
I On C(Σ), `α associates to a marked convex RP2-surfaceΣ ≈ M the Hilbert length of the unique closed geodesichomotopic to α in M.
I (Fricke-Klein ?) The marked length spectrum characterizeshyperbolic structures in F(Σ).
I (Inkang Kim, 2001) The marked Hilbert length spectrum
characterizes RP2-structures in C(Σ).
![Page 150: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/150.jpg)
Fenchel-Nielsen coordinates on the Fricke space F(Σ)
I Cut Σ along N simple closed curves σi into 3-holed spheres(pants). =⇒ Explicit parametrization F(Σ) −→ R6g−6.
l1 l32l
I 2g − 2 = χ(Σ)/χ(P) pants Pj and
N := 3/2(2g − 2) = 3g − 3.
I For a marked hyperbolic surface Σ ≈ M, can represent eachσi by a simple closed geodesic on M. All these σi are disjoint.
![Page 151: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/151.jpg)
Fenchel-Nielsen coordinates on the Fricke space F(Σ)
I Cut Σ along N simple closed curves σi into 3-holed spheres(pants). =⇒ Explicit parametrization F(Σ) −→ R6g−6.
l1 l32l
I 2g − 2 = χ(Σ)/χ(P) pants Pj and
N := 3/2(2g − 2) = 3g − 3.
I For a marked hyperbolic surface Σ ≈ M, can represent eachσi by a simple closed geodesic on M. All these σi are disjoint.
![Page 152: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/152.jpg)
Fenchel-Nielsen coordinates on the Fricke space F(Σ)
I Cut Σ along N simple closed curves σi into 3-holed spheres(pants). =⇒ Explicit parametrization F(Σ) −→ R6g−6.
l1 l32l
I 2g − 2 = χ(Σ)/χ(P) pants Pj and
N := 3/2(2g − 2) = 3g − 3.
I For a marked hyperbolic surface Σ ≈ M, can represent eachσi by a simple closed geodesic on M. All these σi are disjoint.
![Page 153: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/153.jpg)
Fenchel-Nielsen coordinates on the Fricke space F(Σ)
I Cut Σ along N simple closed curves σi into 3-holed spheres(pants). =⇒ Explicit parametrization F(Σ) −→ R6g−6.
l1 l32l
I 2g − 2 = χ(Σ)/χ(P) pants Pj and
N := 3/2(2g − 2) = 3g − 3.
I For a marked hyperbolic surface Σ ≈ M, can represent eachσi by a simple closed geodesic on M. All these σi are disjoint.
![Page 154: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/154.jpg)
Hyperbolic structures on three-holed spheres
I Let li be the length of the geodesic corresponding to σi . Thehyperbolic structure on Pj is completely determined by thethe three lengths of the components of ∂Pj .
I these length functions define a surjection
F(Σ)`−→ RN
+. (1)
which describes the hyperbolic structure on M|σ.
I The components of ∂(M|σ) are identified σ−
i ←→ σ+i , one
pair for each component σi ⊂ σ.
I For each σi , choose τi ∈ R and reidentify M|σ σ−
i ←→ σ+i ,
one pair for each σi , obtaining a new marked hyperbolicsurface
S ≈ Mτ1,...,τN
(Fenchel-Nielsen twists, Thurston earthquakes).
![Page 155: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/155.jpg)
Hyperbolic structures on three-holed spheres
I Let li be the length of the geodesic corresponding to σi . Thehyperbolic structure on Pj is completely determined by thethe three lengths of the components of ∂Pj .
I these length functions define a surjection
F(Σ)`−→ RN
+. (1)
which describes the hyperbolic structure on M|σ.
I The components of ∂(M|σ) are identified σ−
i ←→ σ+i , one
pair for each component σi ⊂ σ.
I For each σi , choose τi ∈ R and reidentify M|σ σ−
i ←→ σ+i ,
one pair for each σi , obtaining a new marked hyperbolicsurface
S ≈ Mτ1,...,τN
(Fenchel-Nielsen twists, Thurston earthquakes).
![Page 156: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/156.jpg)
Hyperbolic structures on three-holed spheres
I Let li be the length of the geodesic corresponding to σi . Thehyperbolic structure on Pj is completely determined by thethe three lengths of the components of ∂Pj .
I these length functions define a surjection
F(Σ)`−→ RN
+. (1)
which describes the hyperbolic structure on M|σ.
I The components of ∂(M|σ) are identified σ−
i ←→ σ+i , one
pair for each component σi ⊂ σ.
I For each σi , choose τi ∈ R and reidentify M|σ σ−
i ←→ σ+i ,
one pair for each σi , obtaining a new marked hyperbolicsurface
S ≈ Mτ1,...,τN
(Fenchel-Nielsen twists, Thurston earthquakes).
![Page 157: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/157.jpg)
Hyperbolic structures on three-holed spheres
I Let li be the length of the geodesic corresponding to σi . Thehyperbolic structure on Pj is completely determined by thethe three lengths of the components of ∂Pj .
I these length functions define a surjection
F(Σ)`−→ RN
+. (1)
which describes the hyperbolic structure on M|σ.
I The components of ∂(M|σ) are identified σ−
i ←→ σ+i , one
pair for each component σi ⊂ σ.
I For each σi , choose τi ∈ R and reidentify M|σ σ−
i ←→ σ+i ,
one pair for each σi , obtaining a new marked hyperbolicsurface
S ≈ Mτ1,...,τN
(Fenchel-Nielsen twists, Thurston earthquakes).
![Page 158: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/158.jpg)
Hyperbolic structures on three-holed spheres
I Let li be the length of the geodesic corresponding to σi . Thehyperbolic structure on Pj is completely determined by thethe three lengths of the components of ∂Pj .
I these length functions define a surjection
F(Σ)`−→ RN
+. (1)
which describes the hyperbolic structure on M|σ.
I The components of ∂(M|σ) are identified σ−
i ←→ σ+i , one
pair for each component σi ⊂ σ.
I For each σi , choose τi ∈ R and reidentify M|σ σ−
i ←→ σ+i ,
one pair for each σi , obtaining a new marked hyperbolicsurface
S ≈ Mτ1,...,τN
(Fenchel-Nielsen twists, Thurston earthquakes).
![Page 159: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/159.jpg)
Fenchel-Nielsen twists (earthquakes)
I Defines an RN -action which is simply transitive on the fibers
of F(Σ)`−→ RN
+.
I (Wolpert 1977) The symplectic form is
N∑
i=1
d`i ∧ dτi
I Completely integrable Hamiltonian system: ` is a Cartesianprojection for symplectomorphism.
F(Σ) −→ RN × RN+
and ` is a moment map for a free, proper HamiltonianRN -action.
![Page 160: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/160.jpg)
Fenchel-Nielsen twists (earthquakes)
I Defines an RN -action which is simply transitive on the fibers
of F(Σ)`−→ RN
+.
I (Wolpert 1977) The symplectic form is
N∑
i=1
d`i ∧ dτi
I Completely integrable Hamiltonian system: ` is a Cartesianprojection for symplectomorphism.
F(Σ) −→ RN × RN+
and ` is a moment map for a free, proper HamiltonianRN -action.
![Page 161: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/161.jpg)
Fenchel-Nielsen twists (earthquakes)
I Defines an RN -action which is simply transitive on the fibers
of F(Σ)`−→ RN
+.
I (Wolpert 1977) The symplectic form is
N∑
i=1
d`i ∧ dτi
I Completely integrable Hamiltonian system: ` is a Cartesianprojection for symplectomorphism.
F(Σ) −→ RN × RN+
and ` is a moment map for a free, proper HamiltonianRN -action.
![Page 162: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/162.jpg)
Fenchel-Nielsen twists (earthquakes)
I Defines an RN -action which is simply transitive on the fibers
of F(Σ)`−→ RN
+.
I (Wolpert 1977) The symplectic form is
N∑
i=1
d`i ∧ dτi
I Completely integrable Hamiltonian system: ` is a Cartesianprojection for symplectomorphism.
F(Σ) −→ RN × RN+
and ` is a moment map for a free, proper HamiltonianRN -action.
![Page 163: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/163.jpg)
Some earthquake deformations in the universal covering
![Page 164: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/164.jpg)
Geometry of C(Σ)
I Hong Chan Kim (1999) generalized Wolpert’s theorem todefine a a symplectomorphism
C(Σ) −→ R16g−6
I ∃ natural completely integrable system in this case?
I (Labourie 1997, Loftin 1999) Mod(Σ)-invariant fibration ofC(Σ) as holomorphic vector bundle over Teichmuller space.
I The fiber over a marked Riemann surface Σ −→ X equalsH0(X ;κ3
X ) comprising holomorphic cubic differentials on X .
I The zero-section corresponds to F(Σ).
![Page 165: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/165.jpg)
Geometry of C(Σ)
I Hong Chan Kim (1999) generalized Wolpert’s theorem todefine a a symplectomorphism
C(Σ) −→ R16g−6
I ∃ natural completely integrable system in this case?
I (Labourie 1997, Loftin 1999) Mod(Σ)-invariant fibration ofC(Σ) as holomorphic vector bundle over Teichmuller space.
I The fiber over a marked Riemann surface Σ −→ X equalsH0(X ;κ3
X ) comprising holomorphic cubic differentials on X .
I The zero-section corresponds to F(Σ).
![Page 166: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/166.jpg)
Geometry of C(Σ)
I Hong Chan Kim (1999) generalized Wolpert’s theorem todefine a a symplectomorphism
C(Σ) −→ R16g−6
I ∃ natural completely integrable system in this case?
I (Labourie 1997, Loftin 1999) Mod(Σ)-invariant fibration ofC(Σ) as holomorphic vector bundle over Teichmuller space.
I The fiber over a marked Riemann surface Σ −→ X equalsH0(X ;κ3
X ) comprising holomorphic cubic differentials on X .
I The zero-section corresponds to F(Σ).
![Page 167: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/167.jpg)
Geometry of C(Σ)
I Hong Chan Kim (1999) generalized Wolpert’s theorem todefine a a symplectomorphism
C(Σ) −→ R16g−6
I ∃ natural completely integrable system in this case?
I (Labourie 1997, Loftin 1999) Mod(Σ)-invariant fibration ofC(Σ) as holomorphic vector bundle over Teichmuller space.
I The fiber over a marked Riemann surface Σ −→ X equalsH0(X ;κ3
X ) comprising holomorphic cubic differentials on X .
I The zero-section corresponds to F(Σ).
![Page 168: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/168.jpg)
Geometry of C(Σ)
I Hong Chan Kim (1999) generalized Wolpert’s theorem todefine a a symplectomorphism
C(Σ) −→ R16g−6
I ∃ natural completely integrable system in this case?
I (Labourie 1997, Loftin 1999) Mod(Σ)-invariant fibration ofC(Σ) as holomorphic vector bundle over Teichmuller space.
I The fiber over a marked Riemann surface Σ −→ X equalsH0(X ;κ3
X ) comprising holomorphic cubic differentials on X .
I The zero-section corresponds to F(Σ).
![Page 169: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/169.jpg)
Geometry of C(Σ)
I Hong Chan Kim (1999) generalized Wolpert’s theorem todefine a a symplectomorphism
C(Σ) −→ R16g−6
I ∃ natural completely integrable system in this case?
I (Labourie 1997, Loftin 1999) Mod(Σ)-invariant fibration ofC(Σ) as holomorphic vector bundle over Teichmuller space.
I The fiber over a marked Riemann surface Σ −→ X equalsH0(X ;κ3
X ) comprising holomorphic cubic differentials on X .
I The zero-section corresponds to F(Σ).
![Page 170: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/170.jpg)
Ingredients of symplectic structure
I Σ oriented closed surface and B Ad-invariant nondegeneatesymmetric pairing on g.
I For ρ ∈ Hom(π,G ), the composition
πρ−→ G
Ad−→ Aut(g)
defines a local coefficient system gAdρ over Σ,
I inheriting a symmetric nondegenerate pairing
gAdρ × gAdρB−→ R
I [ρ] smooth point ⇒ T[ρ]Hom(π,G )/G = H1(Σ, gAdρ).
I Cup-product + coefficient pairing B + orientation =⇒bilinear pairing
H1(Σ, gAdρ)×H1(Σ, gAdρ)ωρ
−→ H2(Σ,R) ∼= R
![Page 171: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/171.jpg)
Ingredients of symplectic structure
I Σ oriented closed surface and B Ad-invariant nondegeneatesymmetric pairing on g.
I For ρ ∈ Hom(π,G ), the composition
πρ−→ G
Ad−→ Aut(g)
defines a local coefficient system gAdρ over Σ,
I inheriting a symmetric nondegenerate pairing
gAdρ × gAdρB−→ R
I [ρ] smooth point ⇒ T[ρ]Hom(π,G )/G = H1(Σ, gAdρ).
I Cup-product + coefficient pairing B + orientation =⇒bilinear pairing
H1(Σ, gAdρ)×H1(Σ, gAdρ)ωρ
−→ H2(Σ,R) ∼= R
![Page 172: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/172.jpg)
Ingredients of symplectic structure
I Σ oriented closed surface and B Ad-invariant nondegeneatesymmetric pairing on g.
I For ρ ∈ Hom(π,G ), the composition
πρ−→ G
Ad−→ Aut(g)
defines a local coefficient system gAdρ over Σ,
I inheriting a symmetric nondegenerate pairing
gAdρ × gAdρB−→ R
I [ρ] smooth point ⇒ T[ρ]Hom(π,G )/G = H1(Σ, gAdρ).
I Cup-product + coefficient pairing B + orientation =⇒bilinear pairing
H1(Σ, gAdρ)×H1(Σ, gAdρ)ωρ
−→ H2(Σ,R) ∼= R
![Page 173: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/173.jpg)
Ingredients of symplectic structure
I Σ oriented closed surface and B Ad-invariant nondegeneatesymmetric pairing on g.
I For ρ ∈ Hom(π,G ), the composition
πρ−→ G
Ad−→ Aut(g)
defines a local coefficient system gAdρ over Σ,
I inheriting a symmetric nondegenerate pairing
gAdρ × gAdρB−→ R
I [ρ] smooth point ⇒ T[ρ]Hom(π,G )/G = H1(Σ, gAdρ).
I Cup-product + coefficient pairing B + orientation =⇒bilinear pairing
H1(Σ, gAdρ)×H1(Σ, gAdρ)ωρ
−→ H2(Σ,R) ∼= R
![Page 174: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/174.jpg)
Ingredients of symplectic structure
I Σ oriented closed surface and B Ad-invariant nondegeneatesymmetric pairing on g.
I For ρ ∈ Hom(π,G ), the composition
πρ−→ G
Ad−→ Aut(g)
defines a local coefficient system gAdρ over Σ,
I inheriting a symmetric nondegenerate pairing
gAdρ × gAdρB−→ R
I [ρ] smooth point ⇒ T[ρ]Hom(π,G )/G = H1(Σ, gAdρ).
I Cup-product + coefficient pairing B + orientation =⇒bilinear pairing
H1(Σ, gAdρ)×H1(Σ, gAdρ)ωρ
−→ H2(Σ,R) ∼= R
![Page 175: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/175.jpg)
Ingredients of symplectic structure
I Σ oriented closed surface and B Ad-invariant nondegeneatesymmetric pairing on g.
I For ρ ∈ Hom(π,G ), the composition
πρ−→ G
Ad−→ Aut(g)
defines a local coefficient system gAdρ over Σ,
I inheriting a symmetric nondegenerate pairing
gAdρ × gAdρB−→ R
I [ρ] smooth point ⇒ T[ρ]Hom(π,G )/G = H1(Σ, gAdρ).
I Cup-product + coefficient pairing B + orientation =⇒bilinear pairing
H1(Σ, gAdρ)×H1(Σ, gAdρ)ωρ
−→ H2(Σ,R) ∼= R
![Page 176: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/176.jpg)
Construction of symplectic structure
I This pairing is skew-symmetric, and hence defines an exterior2-form on the smooth part of Hom(π,G )/G .
I This 2-form is nondegenerate and closed.
I By Ehresmann-Thurston, this induces a symplectic structureon D(G ,X )(Σ).
I On a symplectic manifold (W , ω), functions φ induce vectorfields Ham(φ).
I Both the function φ and the 2-form ω are Ham(φ)-invariant.
![Page 177: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/177.jpg)
Construction of symplectic structure
I This pairing is skew-symmetric, and hence defines an exterior2-form on the smooth part of Hom(π,G )/G .
I This 2-form is nondegenerate and closed.
I By Ehresmann-Thurston, this induces a symplectic structureon D(G ,X )(Σ).
I On a symplectic manifold (W , ω), functions φ induce vectorfields Ham(φ).
I Both the function φ and the 2-form ω are Ham(φ)-invariant.
![Page 178: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/178.jpg)
Construction of symplectic structure
I This pairing is skew-symmetric, and hence defines an exterior2-form on the smooth part of Hom(π,G )/G .
I This 2-form is nondegenerate and closed.
I By Ehresmann-Thurston, this induces a symplectic structureon D(G ,X )(Σ).
I On a symplectic manifold (W , ω), functions φ induce vectorfields Ham(φ).
I Both the function φ and the 2-form ω are Ham(φ)-invariant.
![Page 179: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/179.jpg)
Construction of symplectic structure
I This pairing is skew-symmetric, and hence defines an exterior2-form on the smooth part of Hom(π,G )/G .
I This 2-form is nondegenerate and closed.
I By Ehresmann-Thurston, this induces a symplectic structureon D(G ,X )(Σ).
I On a symplectic manifold (W , ω), functions φ induce vectorfields Ham(φ).
I Both the function φ and the 2-form ω are Ham(φ)-invariant.
![Page 180: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/180.jpg)
Construction of symplectic structure
I This pairing is skew-symmetric, and hence defines an exterior2-form on the smooth part of Hom(π,G )/G .
I This 2-form is nondegenerate and closed.
I By Ehresmann-Thurston, this induces a symplectic structureon D(G ,X )(Σ).
I On a symplectic manifold (W , ω), functions φ induce vectorfields Ham(φ).
I Both the function φ and the 2-form ω are Ham(φ)-invariant.
![Page 181: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/181.jpg)
Construction of symplectic structure
I This pairing is skew-symmetric, and hence defines an exterior2-form on the smooth part of Hom(π,G )/G .
I This 2-form is nondegenerate and closed.
I By Ehresmann-Thurston, this induces a symplectic structureon D(G ,X )(Σ).
I On a symplectic manifold (W , ω), functions φ induce vectorfields Ham(φ).
I Both the function φ and the 2-form ω are Ham(φ)-invariant.
![Page 182: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/182.jpg)
Hamiltonian twist flows on Hom(π, G )
I The Hamiltonian vector field Ham(fα) associated to f and αassigns to a representation ρ in Hom(π,G ) a tangent vector
Ham(fα)[ρ] ∈ T[ρ]Hom(π,G )/G = H1(Σ, gAdρ).
I It is represented by the (Poincare dual) cycle-with-coefficientsupported on α and with coefficient
F (ρ(α)) ∈ gAdρ.
![Page 183: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/183.jpg)
Hamiltonian twist flows on Hom(π, G )
I The Hamiltonian vector field Ham(fα) associated to f and αassigns to a representation ρ in Hom(π,G ) a tangent vector
Ham(fα)[ρ] ∈ T[ρ]Hom(π,G )/G = H1(Σ, gAdρ).
I It is represented by the (Poincare dual) cycle-with-coefficientsupported on α and with coefficient
F (ρ(α)) ∈ gAdρ.
![Page 184: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/184.jpg)
Hamiltonian twist flows on Hom(π, G )
I The Hamiltonian vector field Ham(fα) associated to f and αassigns to a representation ρ in Hom(π,G ) a tangent vector
Ham(fα)[ρ] ∈ T[ρ]Hom(π,G )/G = H1(Σ, gAdρ).
I It is represented by the (Poincare dual) cycle-with-coefficientsupported on α and with coefficient
F (ρ(α)) ∈ gAdρ.
![Page 185: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/185.jpg)
The one-parameter subgroup associated to an invariant
functionI Invariant function
Gf−→ R
and A ∈ G =⇒ one-parameter subgroup
ζ(t) = exp(tF (A)
)∈ G ,
where F (A) ∈ g.I Centralizes A:
ζ(t)Aζ−1 = A
I F (A) is defined by duality:
df (A) ∈ T ∗
AG ∼= g∗B∼= g
I Alternatively, (where X is an arbitrary element of g):
B(F (A),X ) =d
dt
∣∣∣∣t=0
f(A exp(tX )
)
![Page 186: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/186.jpg)
The one-parameter subgroup associated to an invariant
functionI Invariant function
Gf−→ R
and A ∈ G =⇒ one-parameter subgroup
ζ(t) = exp(tF (A)
)∈ G ,
where F (A) ∈ g.I Centralizes A:
ζ(t)Aζ−1 = A
I F (A) is defined by duality:
df (A) ∈ T ∗
AG ∼= g∗B∼= g
I Alternatively, (where X is an arbitrary element of g):
B(F (A),X ) =d
dt
∣∣∣∣t=0
f(A exp(tX )
)
![Page 187: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/187.jpg)
The one-parameter subgroup associated to an invariant
functionI Invariant function
Gf−→ R
and A ∈ G =⇒ one-parameter subgroup
ζ(t) = exp(tF (A)
)∈ G ,
where F (A) ∈ g.I Centralizes A:
ζ(t)Aζ−1 = A
I F (A) is defined by duality:
df (A) ∈ T ∗
AG ∼= g∗B∼= g
I Alternatively, (where X is an arbitrary element of g):
B(F (A),X ) =d
dt
∣∣∣∣t=0
f(A exp(tX )
)
![Page 188: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/188.jpg)
The one-parameter subgroup associated to an invariant
functionI Invariant function
Gf−→ R
and A ∈ G =⇒ one-parameter subgroup
ζ(t) = exp(tF (A)
)∈ G ,
where F (A) ∈ g.I Centralizes A:
ζ(t)Aζ−1 = A
I F (A) is defined by duality:
df (A) ∈ T ∗
AG ∼= g∗B∼= g
I Alternatively, (where X is an arbitrary element of g):
B(F (A),X ) =d
dt
∣∣∣∣t=0
f(A exp(tX )
)
![Page 189: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/189.jpg)
The one-parameter subgroup associated to an invariant
functionI Invariant function
Gf−→ R
and A ∈ G =⇒ one-parameter subgroup
ζ(t) = exp(tF (A)
)∈ G ,
where F (A) ∈ g.I Centralizes A:
ζ(t)Aζ−1 = A
I F (A) is defined by duality:
df (A) ∈ T ∗
AG ∼= g∗B∼= g
I Alternatively, (where X is an arbitrary element of g):
B(F (A),X ) =d
dt
∣∣∣∣t=0
f(A exp(tX )
)
![Page 190: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/190.jpg)
Generalized twist deformations
I When α is a simple closed curve, then a flow Φt onHom(π,G ) exists, which covers the (local) flow of theHamiltonian vector field Ham(fα).
I When α is, for example, the nonseparating curve A1 in thestandard presentation
π = 〈A1,B1, . . . ,Ag ,Bg | A1B1A−11 B−1
1 . . . ,AgBgA−1g B−1
g = 1〉
this flow has the following description in terms of generators:.
I Φt(γ) = ρ(γ) is constant if γ is either Ai for 1 ≤ i ≤ g or Bi
for 2 ≤ i ≤ g .
I Φt(B1) = ρ(B1)ζ(t).
I Similar construction when γ separates..
![Page 191: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/191.jpg)
Generalized twist deformations
I When α is a simple closed curve, then a flow Φt onHom(π,G ) exists, which covers the (local) flow of theHamiltonian vector field Ham(fα).
I When α is, for example, the nonseparating curve A1 in thestandard presentation
π = 〈A1,B1, . . . ,Ag ,Bg | A1B1A−11 B−1
1 . . . ,AgBgA−1g B−1
g = 1〉
this flow has the following description in terms of generators:.
I Φt(γ) = ρ(γ) is constant if γ is either Ai for 1 ≤ i ≤ g or Bi
for 2 ≤ i ≤ g .
I Φt(B1) = ρ(B1)ζ(t).
I Similar construction when γ separates..
![Page 192: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/192.jpg)
Generalized twist deformations
I When α is a simple closed curve, then a flow Φt onHom(π,G ) exists, which covers the (local) flow of theHamiltonian vector field Ham(fα).
I When α is, for example, the nonseparating curve A1 in thestandard presentation
π = 〈A1,B1, . . . ,Ag ,Bg | A1B1A−11 B−1
1 . . . ,AgBgA−1g B−1
g = 1〉
this flow has the following description in terms of generators:.
I Φt(γ) = ρ(γ) is constant if γ is either Ai for 1 ≤ i ≤ g or Bi
for 2 ≤ i ≤ g .
I Φt(B1) = ρ(B1)ζ(t).
I Similar construction when γ separates..
![Page 193: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/193.jpg)
Generalized twist deformations
I When α is a simple closed curve, then a flow Φt onHom(π,G ) exists, which covers the (local) flow of theHamiltonian vector field Ham(fα).
I When α is, for example, the nonseparating curve A1 in thestandard presentation
π = 〈A1,B1, . . . ,Ag ,Bg | A1B1A−11 B−1
1 . . . ,AgBgA−1g B−1
g = 1〉
this flow has the following description in terms of generators:.
I Φt(γ) = ρ(γ) is constant if γ is either Ai for 1 ≤ i ≤ g or Bi
for 2 ≤ i ≤ g .
I Φt(B1) = ρ(B1)ζ(t).
I Similar construction when γ separates..
![Page 194: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/194.jpg)
Generalized twist deformations
I When α is a simple closed curve, then a flow Φt onHom(π,G ) exists, which covers the (local) flow of theHamiltonian vector field Ham(fα).
I When α is, for example, the nonseparating curve A1 in thestandard presentation
π = 〈A1,B1, . . . ,Ag ,Bg | A1B1A−11 B−1
1 . . . ,AgBgA−1g B−1
g = 1〉
this flow has the following description in terms of generators:.
I Φt(γ) = ρ(γ) is constant if γ is either Ai for 1 ≤ i ≤ g or Bi
for 2 ≤ i ≤ g .
I Φt(B1) = ρ(B1)ζ(t).
I Similar construction when γ separates..
![Page 195: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/195.jpg)
Generalized twist deformations
I When α is a simple closed curve, then a flow Φt onHom(π,G ) exists, which covers the (local) flow of theHamiltonian vector field Ham(fα).
I When α is, for example, the nonseparating curve A1 in thestandard presentation
π = 〈A1,B1, . . . ,Ag ,Bg | A1B1A−11 B−1
1 . . . ,AgBgA−1g B−1
g = 1〉
this flow has the following description in terms of generators:.
I Φt(γ) = ρ(γ) is constant if γ is either Ai for 1 ≤ i ≤ g or Bi
for 2 ≤ i ≤ g .
I Φt(B1) = ρ(B1)ζ(t).
I Similar construction when γ separates..
![Page 196: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/196.jpg)
Twist and bulging deformations for RP2-structures
I Apply the previous general construction to G = SL(3,R) andthe two invariant functions `, β defined earlier:
λ1 0 00 λ2 00 0 λ3
(`,β)−−−→
(log(λ1)− log(λ3)
log(λ2)
)
I The corresponding one-parameter subgroups in PGL(3,R) are:
ζ`(t) :=
et 0 00 1 00 0 e−t
, ζβ(t) := e−t/3
1 0 00 et 00 0 1
![Page 197: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/197.jpg)
Twist and bulging deformations for RP2-structures
I Apply the previous general construction to G = SL(3,R) andthe two invariant functions `, β defined earlier:
λ1 0 00 λ2 00 0 λ3
(`,β)−−−→
(log(λ1)− log(λ3)
log(λ2)
)
I The corresponding one-parameter subgroups in PGL(3,R) are:
ζ`(t) :=
et 0 00 1 00 0 e−t
, ζβ(t) := e−t/3
1 0 00 et 00 0 1
![Page 198: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/198.jpg)
Twist and bulging deformations for RP2-structures
I Apply the previous general construction to G = SL(3,R) andthe two invariant functions `, β defined earlier:
λ1 0 00 λ2 00 0 λ3
(`,β)−−−→
(log(λ1)− log(λ3)
log(λ2)
)
I The corresponding one-parameter subgroups in PGL(3,R) are:
ζ`(t) :=
et 0 00 1 00 0 e−t
, ζβ(t) := e−t/3
1 0 00 et 00 0 1
![Page 199: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/199.jpg)
Bulging conics along a triangle in RP2
I When applied to a hyperbolic structure, the flow of Ham(`α)is just the ordinary Fenchel-Nielsen earthquake deformationand the developing image Ω is unchanged.
I However, the flow of Ham(βα) changes Ω by bulging it alonga triangle tangent to ∂Ω.
![Page 200: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/200.jpg)
Bulging conics along a triangle in RP2
I When applied to a hyperbolic structure, the flow of Ham(`α)is just the ordinary Fenchel-Nielsen earthquake deformationand the developing image Ω is unchanged.
I However, the flow of Ham(βα) changes Ω by bulging it alonga triangle tangent to ∂Ω.
![Page 201: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/201.jpg)
Bulging conics along a triangle in RP2
I When applied to a hyperbolic structure, the flow of Ham(`α)is just the ordinary Fenchel-Nielsen earthquake deformationand the developing image Ω is unchanged.
I However, the flow of Ham(βα) changes Ω by bulging it alonga triangle tangent to ∂Ω.
![Page 202: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/202.jpg)
Bulging domains in RP2
I Start with a properly domain Ω whose boundary ∂Ω is strictlyconvex and C 1. (For example, ∂Ω a conic.) Each geodesicembeds in a triangle tangent to ∂Ω.
I Choose a collection Λ of disjoint lines in Ω, with instructionshow to deform along Λ (for each λ ∈ Λ, a one-parametersubgroup of SL(3,R) preserving λ.
I Fixing a basepoint in the complement of Λ, bulge/earthquakethe curve inside the triangles tangent to ∂Ω.
I Obtain a sequence of piecewise conics converging to the limitcurve.
![Page 203: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/203.jpg)
Bulging domains in RP2
I Start with a properly domain Ω whose boundary ∂Ω is strictlyconvex and C 1. (For example, ∂Ω a conic.) Each geodesicembeds in a triangle tangent to ∂Ω.
I Choose a collection Λ of disjoint lines in Ω, with instructionshow to deform along Λ (for each λ ∈ Λ, a one-parametersubgroup of SL(3,R) preserving λ.
I Fixing a basepoint in the complement of Λ, bulge/earthquakethe curve inside the triangles tangent to ∂Ω.
I Obtain a sequence of piecewise conics converging to the limitcurve.
![Page 204: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/204.jpg)
Bulging domains in RP2
I Start with a properly domain Ω whose boundary ∂Ω is strictlyconvex and C 1. (For example, ∂Ω a conic.) Each geodesicembeds in a triangle tangent to ∂Ω.
I Choose a collection Λ of disjoint lines in Ω, with instructionshow to deform along Λ (for each λ ∈ Λ, a one-parametersubgroup of SL(3,R) preserving λ.
I Fixing a basepoint in the complement of Λ, bulge/earthquakethe curve inside the triangles tangent to ∂Ω.
I Obtain a sequence of piecewise conics converging to the limitcurve.
![Page 205: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/205.jpg)
Bulging domains in RP2
I Start with a properly domain Ω whose boundary ∂Ω is strictlyconvex and C 1. (For example, ∂Ω a conic.) Each geodesicembeds in a triangle tangent to ∂Ω.
I Choose a collection Λ of disjoint lines in Ω, with instructionshow to deform along Λ (for each λ ∈ Λ, a one-parametersubgroup of SL(3,R) preserving λ.
I Fixing a basepoint in the complement of Λ, bulge/earthquakethe curve inside the triangles tangent to ∂Ω.
I Obtain a sequence of piecewise conics converging to the limitcurve.
![Page 206: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/206.jpg)
Bulging domains in RP2
I Start with a properly domain Ω whose boundary ∂Ω is strictlyconvex and C 1. (For example, ∂Ω a conic.) Each geodesicembeds in a triangle tangent to ∂Ω.
I Choose a collection Λ of disjoint lines in Ω, with instructionshow to deform along Λ (for each λ ∈ Λ, a one-parametersubgroup of SL(3,R) preserving λ.
I Fixing a basepoint in the complement of Λ, bulge/earthquakethe curve inside the triangles tangent to ∂Ω.
I Obtain a sequence of piecewise conics converging to the limitcurve.
![Page 207: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/207.jpg)
A domain in RP2 covering a closed surface
![Page 208: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/208.jpg)
Iterated bulging of convex domains in RP2: Speculation
I If Ω covers a closed convex RP2-surface with χ < 0, then ∂Ωis obtained from a conic by iterated bulgings and earthquakes.
I Is every properly convex domain Ω ⊂ RP2 with strictly convexC 1 boundary obtained by iterated bulging-earthquaking?
I Thurston proved that any two marked hyperbolic structureson Σ can be related by (left)-earthquake along a uniquemeasured geodesic lamination. Generalize this to convexRP2-structures.
![Page 209: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/209.jpg)
Iterated bulging of convex domains in RP2: Speculation
I If Ω covers a closed convex RP2-surface with χ < 0, then ∂Ωis obtained from a conic by iterated bulgings and earthquakes.
I Is every properly convex domain Ω ⊂ RP2 with strictly convexC 1 boundary obtained by iterated bulging-earthquaking?
I Thurston proved that any two marked hyperbolic structureson Σ can be related by (left)-earthquake along a uniquemeasured geodesic lamination. Generalize this to convexRP2-structures.
![Page 210: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/210.jpg)
Iterated bulging of convex domains in RP2: Speculation
I If Ω covers a closed convex RP2-surface with χ < 0, then ∂Ωis obtained from a conic by iterated bulgings and earthquakes.
I Is every properly convex domain Ω ⊂ RP2 with strictly convexC 1 boundary obtained by iterated bulging-earthquaking?
I Thurston proved that any two marked hyperbolic structureson Σ can be related by (left)-earthquake along a uniquemeasured geodesic lamination. Generalize this to convexRP2-structures.
![Page 211: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/211.jpg)
Iterated bulging of convex domains in RP2: Speculation
I If Ω covers a closed convex RP2-surface with χ < 0, then ∂Ωis obtained from a conic by iterated bulgings and earthquakes.
I Is every properly convex domain Ω ⊂ RP2 with strictly convexC 1 boundary obtained by iterated bulging-earthquaking?
I Thurston proved that any two marked hyperbolic structureson Σ can be related by (left)-earthquake along a uniquemeasured geodesic lamination. Generalize this to convexRP2-structures.
![Page 212: Deformations of geometric structures and representations ...wmg/kaist.pdf · Geometry through symmetry I In his 1872 Erlangen Program, Felix Klein proposed that a geometry is the](https://reader031.vdocuments.mx/reader031/viewer/2022020114/5b949d1d09d3f2130d8d0ec2/html5/thumbnails/212.jpg)