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CP1-structures and dynamics in moduli space
Subhojoy GuptaCenter for Quantum Geometry
of Moduli Spaces (QGM), Aarhus
Rolf Nevanlinna ColloquiumHelsinki
August 8, 2013
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partly joint work with Shinpei Baba
1 CP1-structuresBackgroundThe density result
2 Dynamics in Mg
Grafting raysTeichmuller raysThe asymptoticity result
3 Idea of the proofWarm-up caseGeneral case
4 Further questions
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CP1-structuresBackground
Definition
Let S be a closed oriented surface of genus g ≥ 2.
Definition
A complex projective structure on S is a maximal atlas of charts to CP1
such that transition maps are restrictions of elements of Aut(CP1) =PSL2(C).
ψ ◦ φ−1 is a Mobius map z 7→ az+bcz+d
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CP1-structuresBackground
CP1 structure: a “global” definition
A complex projective structure is specified by:
A developing map that is an immersion from the universal coverf : S → CP1.
A holonomy representation ρ : π1(S)→ PSL2(C) that is compatible:
f ◦ γ = ρ(γ) ◦ f for all γ ∈ π1(S).
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CP1-structuresBackground
Examples: hyperbolic structures
Definition
A hyperbolic structure on S is a collection of charts to the hyperbolicplane H2 with transition maps in PSL2(R) = Isom+(H2).
This is a special case of a complex-projective structure:
Can identify H2 with the upper hemisphere on CP1.
The holonomy representation is Fuchsian, PSL2(R) ↪→ PSL2(C).
Uniformization theorem: any Riemann surface has a hyperbolic structure.
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CP1-structuresBackground
Other examples
“Boundary at infinity” of hyperbolic 3-manifolds.
e.g. quasifuchsian:
Others with indiscrete holonomy.
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CP1-structuresBackground
A bundle picture
A CP1-structure defines a complex structure on S since the transitionmaps are conformal.
Pg = {space of marked CP1-structures on S}
Pg
Tg
Mg
p
π
where Tg is Teichmuller space and Mg is the moduli space of Riemannsurfaces.
p−1(X ) ∼= Q(X ) = {holomorphic quadratic differentials on X}
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CP1-structuresBackground
A bundle picture
A CP1-structure defines a complex structure on S since the transitionmaps are conformal.
Pg = {space of marked CP1-structures on S}
Pg
Tg
Mg
p
π
where Tg is Teichmuller space and Mg is the moduli space of Riemannsurfaces.
p−1(X ) ∼= Q(X ) = {holomorphic quadratic differentials on X}
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CP1-structuresBackground
The holonomy map
Pg
Tg
Mg
χ(S)hol
p
π
χ(S): the PSL2(C)-character variety.
Goal: Let ρ ∈ χ(S). Describe the holonomy level sets Pρ = hol−1(ρ).
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CP1-structuresBackground
The holonomy map
Pg
Tg
Mg
χ(S)hol
p
π
χ(S): the PSL2(C)-character variety.
Goal: Let ρ ∈ χ(S). Describe the holonomy level sets Pρ = hol−1(ρ).
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CP1-structuresThe density result
The holonomy map
Pg χ(S)hol
hol is a local homeomorphism, but not a covering map. (Hejhal,Earle, Hubbard)
The image of hol is almost all of one of the two irreduciblecomponents. (Gallo-Kapovich-Marden)
For ρ quasifuchsian, Pρ is infinite. (Goldman)
If Pρ 6= ∅, it is infinite. (Gallo-Kapovich-Marden, Baba)
Theorem (Baba-G.)
If Pρ 6= ∅ the projection of Pρ to Mg is dense.
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CP1-structuresThe density result
The holonomy map
Pg χ(S)hol
hol is a local homeomorphism, but not a covering map. (Hejhal,Earle, Hubbard)
The image of hol is almost all of one of the two irreduciblecomponents. (Gallo-Kapovich-Marden)
For ρ quasifuchsian, Pρ is infinite. (Goldman)
If Pρ 6= ∅, it is infinite. (Gallo-Kapovich-Marden, Baba)
Theorem (Baba-G.)
If Pρ 6= ∅ the projection of Pρ to Mg is dense.
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CP1-structuresThe density result
The holonomy map
Pg χ(S)hol
hol is a local homeomorphism, but not a covering map. (Hejhal,Earle, Hubbard)
The image of hol is almost all of one of the two irreduciblecomponents. (Gallo-Kapovich-Marden)
For ρ quasifuchsian, Pρ is infinite. (Goldman)
If Pρ 6= ∅, it is infinite. (Gallo-Kapovich-Marden, Baba)
Theorem (Baba-G.)
If Pρ 6= ∅ the projection of Pρ to Mg is dense.
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CP1-structuresThe density result
The holonomy map
Pg χ(S)hol
hol is a local homeomorphism, but not a covering map. (Hejhal,Earle, Hubbard)
The image of hol is almost all of one of the two irreduciblecomponents. (Gallo-Kapovich-Marden)
For ρ quasifuchsian, Pρ is infinite. (Goldman)
If Pρ 6= ∅, it is infinite. (Gallo-Kapovich-Marden, Baba)
Theorem (Baba-G.)
If Pρ 6= ∅ the projection of Pρ to Mg is dense.
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CP1-structuresThe density result
The density result
Obtained by relating the dynamics of grafting rays and Teichmuller rays.
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CP1-structuresThe density result
The density result
Obtained by relating the dynamics of grafting rays and Teichmuller rays.
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Dynamics in Mg
Grafting rays
Projective grafting gives deformations of CP1-structures from ahyperbolic one. Grafting rays are the shadows of these deformations inTeichmuller space:
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Dynamics in Mg
Grafting rays
Projective grafting
Given a hyperbolic surface X and a simple closed curve γ, one candeform by bending :
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Dynamics in Mg
Grafting rays
Projective grafting
Given a hyperbolic surface X and a simple closed curve γ, one candeform by bending :
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Dynamics in Mg
Grafting rays
Projective grafting
Given a hyperbolic surface X and a simple closed curve γ, one candeform by bending :
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Dynamics in Mg
Grafting rays
Projective grafting
Given a hyperbolic surface X and a simple closed curve γ, one candeform by bending :
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Dynamics in Mg
Grafting rays
Projective grafting
Given a hyperbolic surface X and a simple closed curve γ, one candeform by bending :
2π-grafting along a multicurve preserves Fuchsian holonomy (Goldman)
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Dynamics in Mg
Grafting rays
2π-grafting
More generally, 2π-grafting along an admissible curve preserves theholonomy.
admissible: developing image embeds,loxodromic holonomy
Baba: existence of admissible curves.
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Dynamics in Mg
Grafting rays
Projective grafting
Can graft along weighted multicurves:
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Dynamics in Mg
Grafting rays
Projective grafting
Can graft along weighted multicurves:
... and measured laminations.
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Dynamics in Mg
Grafting rays
Measured geodesic laminations
A measured geodesic lamination λ is a closed set on a hyperbolic surfacewhich is a union of a disjoint collection of simple geodesics, equippedwith a transverse measure µ .
Weighted multicurves are dense in ML.
(Thurston parametrization) Projective grafting along measuredlaminations yields:
Tg ×ML ∼= Pg
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Dynamics in Mg
Grafting rays
Measured geodesic laminations
A measured geodesic lamination λ is a closed set on a hyperbolic surfacewhich is a union of a disjoint collection of simple geodesics, equippedwith a transverse measure µ .
Weighted multicurves are dense in ML.
(Thurston parametrization) Projective grafting along measuredlaminations yields:
Tg ×ML ∼= Pg
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Dynamics in Mg
Grafting rays
Measured geodesic laminations
A measured geodesic lamination λ is a closed set on a hyperbolic surfacewhich is a union of a disjoint collection of simple geodesics, equippedwith a transverse measure µ .
Weighted multicurves are dense in ML.
(Thurston parametrization) Projective grafting along measuredlaminations yields:
Tg ×ML ∼= Pg
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Dynamics in Mg
Grafting rays
Grafting rays
Given a hyperbolic surface X , can graft along a measured geodesiclamination and its rescalings tλ. Grafting rays are the shadows of thesedeformations in Teichmuller space:
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Dynamics in Mg
Grafting rays
Grafting rays
X 7→ grtλX
For a simple closed curve:
More generally:
widens transverse to the lamination
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Dynamics in Mg
Grafting rays
Grafting rays
X 7→ grtλX
For a simple closed curve:
More generally:
widens transverse to the lamination
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Dynamics in Mg
Teichmuller rays
Teichmuller metric
For X ,Y ∈ Tg we can define the Teichmuller distance
dT (X ,Y ) =1
2inff
ln K
where K is the dilatation of a quasiconformal map f : X → Y .
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Dynamics in Mg
Teichmuller rays
Teichmuller metric
For X ,Y ∈ Tg we can define the Teichmuller distance
dT (X ,Y ) =1
2inff
ln K
where K is the dilatation of a quasiconformal map f : X → Y .
Toy example: Rectangles R1 and R2 of different moduli.
dT (R1,R2) = 12 ln L, realized by the stretch map.
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Dynamics in Mg
Teichmuller rays
Teichmuller metric
For X ,Y ∈ Tg we can define the Teichmuller distance
dT (X ,Y ) =1
2inff
ln K
where K is the dilatation of a quasiconformal map f : X → Y .
Facts:
dT is a complete metric.
dT is the Finsler metric given by the L1-norm on Q(X ).
Co-tangent space T ∗XTg ∼= Q(X ).
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Dynamics in Mg
Teichmuller rays
Teichmuller rays
A Teichmuller ray with basepoint X in the direction of q ∈ Q(X ):
q determines a singular flat metric |q(z)||dz |2 on X together with avertical measured foliation Fv and a horizontal measured foliation Fh.
Stretch in the horizontal direction by a factor of e2t .
This ray is geodesic in the Teichmuller metric.
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Dynamics in Mg
Teichmuller rays
Teichmuller rays
A Teichmuller ray with basepoint X in the direction of q ∈ Q(X ):
q determines a singular flat metric |q(z)||dz |2 on X together with avertical measured foliation Fv and a horizontal measured foliation Fh.
Stretch in the horizontal direction by a factor of e2t .
This ray is geodesic in the Teichmuller metric.
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Dynamics in Mg
Teichmuller rays
Teichmuller rays
A Teichmuller ray with basepoint X in the direction of q ∈ Q(X ):
q determines a singular flat metric |q(z)||dz |2 on X together with avertical measured foliation Fv and a horizontal measured foliation Fh.
Stretch in the horizontal direction by a factor of e2t .
This ray is geodesic in the Teichmuller metric.
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Dynamics in Mg
Teichmuller rays
Teichmuller rays
A Teichmuller ray with basepoint X in the direction of λ ∈ML:
(X , q)←→ (X ,Fv )←→ (X , λ)
Q(X ) ∼=MF via the map q 7→ Fv (q). (Hubbard-Masur)
MF ∼=ML.
Fact
For any X ∈ Tg , and almost every λ ∈ML the Teichmuller rayTeichtλ(X ) is dense in Mg .
Follows from the ergodicity of the Teichmuller geodesic flow.(Masur, Veech)
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Dynamics in Mg
Teichmuller rays
Teichmuller rays
A Teichmuller ray with basepoint X in the direction of λ ∈ML:
(X , q)←→ (X ,Fv )←→ (X , λ)
Q(X ) ∼=MF via the map q 7→ Fv (q). (Hubbard-Masur)
MF ∼=ML.
Fact
For any X ∈ Tg , and almost every λ ∈ML the Teichmuller rayTeichtλ(X ) is dense in Mg .
Follows from the ergodicity of the Teichmuller geodesic flow.(Masur, Veech)
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Dynamics in Mg
Teichmuller rays
Teichmuller rays
A Teichmuller ray with basepoint X in the direction of λ ∈ML:
(X , q)←→ (X ,Fv )←→ (X , λ)
Q(X ) ∼=MF via the map q 7→ Fv (q). (Hubbard-Masur)
MF ∼=ML.
Fact
For any X ∈ Tg , and almost every λ ∈ML the Teichmuller rayTeichtλ(X ) is dense in Mg .
Follows from the ergodicity of the Teichmuller geodesic flow.(Masur, Veech)
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Dynamics in Mg
Teichmuller rays
Comparison of grafting and Teichmuller rays:
Diaz-Kim,Choi-Dumas-Rafi
(“fellow-traveling” results)
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Dynamics in Mg
The asymptoticity result
The asymptoticity result
Theorem (G.)
Let (X , λ) ∈ Tg ×ML. Then there exists a Y ∈ Tg such that thegrafting ray determined by (X , λ) is strongly asymptotic to theTeichmuller ray determined by (Y , λ), that is,
dT (gre2tλX ,TeichtλY )→ 0
as t →∞.
Corollary
Almost every grafting ray projects to a dense set in moduli space Mg .
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Dynamics in Mg
The asymptoticity result
The asymptoticity result
Theorem (G.)
Let (X , λ) ∈ Tg ×ML. Then there exists a Y ∈ Tg such that thegrafting ray determined by (X , λ) is strongly asymptotic to theTeichmuller ray determined by (Y , λ), that is,
dT (gre2tλX ,TeichtλY )→ 0
as t →∞.
Corollary
Almost every grafting ray projects to a dense set in moduli space Mg .
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Dynamics in Mg
The asymptoticity result
Density of 2π-graftings
Theorem (Baba-G.)
Let C ∈ Pg . Then the set
S = {gr2πγC | γ is an admissible multicurve on C }
projects to a dense set in moduli space Mg .
Corollary
Any non-empty holonomy level set is dense in moduli space.
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Dynamics in Mg
The asymptoticity result
Density of 2π-graftings
Theorem (Baba-G.)
Let C ∈ Pg . Then the set
S = {gr2πγC | γ is an admissible multicurve on C }
projects to a dense set in moduli space Mg .
Corollary
Any non-empty holonomy level set is dense in moduli space.
![Page 46: CP1-structures and dynamics in moduli spacemath.iisc.ernet.in/~subhojoy/public_html/About_Me_files/Helsinki4.pdf · CP1-structures and dynamics in moduli space Subhojoy Gupta Center](https://reader036.vdocuments.us/reader036/viewer/2022062602/5ebafc56fe245046654d8100/html5/thumbnails/46.jpg)
Idea of the proof
Warm-up case
Idea of proof of asymptoticity (multicurve case)
Step 1. Consider the conformal limit of the grafting ray.
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Idea of the proof
Warm-up case
Idea of proof of asymptoticity (multicurve case)
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Idea of the proof
Warm-up case
Idea of proof of asymptoticity (multicurve case)
Step 2. Obtain Y∞ by specifying a meromorphic quadratic differentialwith poles of order 2. (Strebel)
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Idea of the proof
Warm-up case
Theorem of Strebel
Theorem
Let Σ be a Riemann surface of genus g ≥ 2, and P a collection of n ≥ 1points, and a1, a2, . . . , an a tuple of positive reals.Then there exists a meromorphic quadratic differential q on Σ with
poles at P of order 2 and residues a1, . . . an
all vertical leaves (except the critical trajectories) are closed andfoliate punctured disks around P.
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Idea of the proof
Warm-up case
Idea of proof of asymptoticity (multicurve case)
Step 2. Obtain Y∞ by specifying a meromorphic quadratic differentialwith poles of order 2. (Strebel)
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Idea of the proof
Warm-up case
Idea of proof of asymptoticity (multicurve case)
Step 3. Adjust g to an almost-conformal map that takes circles tocircles. (Uses standard distortion theorems.)
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Idea of the proof
Warm-up case
Idea of proof of asymptoticity (multicurve case)
Step 4. Truncating along those circles and gluing gives the requiredmap.
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Idea of the proof
Warm-up case
Quasiconformal interpolation
Lemma
Let f : D→ C be a univalent conformal map such that f (0) = 0 andf ′(0) = 1. For any ε > 0 there exists an 0 < r < 1 and a mapF : D→ f (D) such that
F is (1 + ε)-quasiconformal
F restricts to the identity map on Br
F |∂D = f |∂D.
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Idea of the proof
General case
Warm-up II: the maximal case
λ is a maximal lamination with complement a union of ideal triangles.
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Idea of the proof
General case
Warm-up II: the maximal case
λ is a maximal lamination with complement a union of ideal triangles.
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Idea of the proof
General case
Warm-up II: the maximal case
λ is a maximal lamination with complement a union of ideal triangles.
grafting limit
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Idea of the proof
General case
Warm-up II: the maximal case
(Step 2) Corresponding singular-flat limit of the Teichmuller ray:
zdz2
4g − 4 copies of (C, zdz2).
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Idea of the proof
General case
Warm-up II: the maximal case
(Steps 3 and 4) Adjust, and truncate and glue
to get an almost-conformal map between surfaces along the rays.
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Idea of the proof
General case
General case
conformal limit of grafting ray:
Corresponding limit of the Teichmuller ray is given by a meromorphicquadratic differential with a “half-plane structure”.
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Idea of the proof
General case
General case
conformal limit of grafting ray:
Corresponding limit of the Teichmuller ray is given by a meromorphicquadratic differential with a “half-plane structure”.
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Idea of the proof
General case
General case
conformal limit of grafting ray:
Corresponding limit of the Teichmuller ray is given by a meromorphicquadratic differential with a “half-plane structure”.
![Page 62: CP1-structures and dynamics in moduli spacemath.iisc.ernet.in/~subhojoy/public_html/About_Me_files/Helsinki4.pdf · CP1-structures and dynamics in moduli space Subhojoy Gupta Center](https://reader036.vdocuments.us/reader036/viewer/2022062602/5ebafc56fe245046654d8100/html5/thumbnails/62.jpg)
Idea of the proof
General case
A generalization of Strebel’s result
Half-plane differential: A meromorphic quadratic differential with poles oforder ≥ 4 and a connected critical graph.
Theorem (G.)
Given any Riemann surface Σ and a set of points P on it there exists ahalf-plane differential with poles at P with prescribed local data.
local data: order, residue, and coefficient of the leading order term.
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Further questions
Further questions
Extend to the case of complex earthquakes and Teichmuller disks.
Motivating analogy:Teichmuller horocycle flow ←→ Earthquake flow (Mirzakhani)Teichmuller geodesic flow ←→ ?
Measure-theoretic questions. E.g. Do the 2π-graftings equidistributein Mg?
Mapping class group action on the character variety χ(S). E.g.Which orbits are dense in the complement of quasifuchsian space?
Pg χ(S)hol
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Further questions
Further questions
Extend to the case of complex earthquakes and Teichmuller disks.
Motivating analogy:Teichmuller horocycle flow ←→ Earthquake flow (Mirzakhani)Teichmuller geodesic flow ←→ ?
Measure-theoretic questions. E.g. Do the 2π-graftings equidistributein Mg?
Mapping class group action on the character variety χ(S). E.g.Which orbits are dense in the complement of quasifuchsian space?
Pg χ(S)hol
![Page 65: CP1-structures and dynamics in moduli spacemath.iisc.ernet.in/~subhojoy/public_html/About_Me_files/Helsinki4.pdf · CP1-structures and dynamics in moduli space Subhojoy Gupta Center](https://reader036.vdocuments.us/reader036/viewer/2022062602/5ebafc56fe245046654d8100/html5/thumbnails/65.jpg)
Further questions
Further questions
Extend to the case of complex earthquakes and Teichmuller disks.
Motivating analogy:Teichmuller horocycle flow ←→ Earthquake flow (Mirzakhani)Teichmuller geodesic flow ←→ ?
Measure-theoretic questions. E.g. Do the 2π-graftings equidistributein Mg?
Mapping class group action on the character variety χ(S). E.g.Which orbits are dense in the complement of quasifuchsian space?
Pg χ(S)hol
![Page 66: CP1-structures and dynamics in moduli spacemath.iisc.ernet.in/~subhojoy/public_html/About_Me_files/Helsinki4.pdf · CP1-structures and dynamics in moduli space Subhojoy Gupta Center](https://reader036.vdocuments.us/reader036/viewer/2022062602/5ebafc56fe245046654d8100/html5/thumbnails/66.jpg)
Further questions
Further questions
Extend to the case of complex earthquakes and Teichmuller disks.
Motivating analogy:Teichmuller horocycle flow ←→ Earthquake flow (Mirzakhani)Teichmuller geodesic flow ←→ ?
Measure-theoretic questions. E.g. Do the 2π-graftings equidistributein Mg?
Mapping class group action on the character variety χ(S). E.g.Which orbits are dense in the complement of quasifuchsian space?
Pg χ(S)hol