On Convex Projective Structures: Part I

Alex Casella——————————————————————————

5th Australian Mathematical SciencesStudent Conference

——————————————————————————Joint work with D. Tate and S. Tillmann

7th December 2017

Alex Casella On Convex Projective Structures: Part I 7th December 2017 1 / 16

Motivation

The goal of Topology is to understand topological spaces, classifythem or find ways to describe them.

To understand a topological space, one can look at its properties.

Question

Does my topological space admit any particular geometry? Like anice Riemannian metric?

Closed Orientable Surfaces:Sphere: spherical geometryTorus: Euclidean geometryOtherwise: Hyperbolic geometry

Alex Casella On Convex Projective Structures: Part I 7th December 2017 2 / 16

Motivation

The goal of Topology is to understand topological spaces, classifythem or find ways to describe them.

To understand a topological space, one can look at its properties.

Question

Does my topological space admit any particular geometry? Like anice Riemannian metric?

Closed Orientable Surfaces:Sphere: spherical geometryTorus: Euclidean geometryOtherwise: Hyperbolic geometry

Alex Casella On Convex Projective Structures: Part I 7th December 2017 2 / 16

Motivation

The goal of Topology is to understand topological spaces, classifythem or find ways to describe them.

To understand a topological space, one can look at its properties.

Question

Does my topological space admit any particular geometry? Like anice Riemannian metric?

Closed Orientable Surfaces:Sphere: spherical geometryTorus: Euclidean geometryOtherwise: Hyperbolic geometry

Alex Casella On Convex Projective Structures: Part I 7th December 2017 2 / 16

Convex Projective Structures

Convex Projective Structures

A convex projective structure of a surface S is a pair (Ω,Γ):

i) Ω ⊂ RP2 properly convex domain

ii) Γ ≤ PGL3(R) discrete subgrouppreserving Ω

iii) Ω/Γ ∼= S

Alex Casella On Convex Projective Structures: Part I 7th December 2017 3 / 16

Convex Projective Structures

Convex Projective Structures

A convex projective structure of a surface S is a pair (Ω,Γ):

i) Ω ⊂ RP2 properly convex domain

ii) Γ ≤ PGL3(R) discrete subgrouppreserving Ω

iii) Ω/Γ ∼= S

Alex Casella On Convex Projective Structures: Part I 7th December 2017 3 / 16

Convex Projective Structures

Convex Projective Structures

A convex projective structure of a surface S is a pair (Ω,Γ):

i) Ω ⊂ RP2 properly convex domain

ii) Γ ≤ PGL3(R) discrete subgrouppreserving Ω

iii) Ω/Γ ∼= S

Alex Casella On Convex Projective Structures: Part I 7th December 2017 3 / 16

Convex Projective Structures

Convex Projective Structures

A convex projective structure of a surface S is a pair (Ω,Γ):i) Ω ⊂ RP2 properly convex domainii) Γ ≤ PGL3(R) discrete subgroupiii) Ω/Γ ∼= S

Alex Casella On Convex Projective Structures: Part I 7th December 2017 4 / 16

Convex Projective Structures

Convex Projective Structures

A convex projective structure of a surface S is a pair (Ω,Γ):i) Ω ⊂ RP2 properly convex domainii) Γ ≤ PGL3(R) discrete subgroupiii) Ω/Γ ∼= S

Alex Casella On Convex Projective Structures: Part I 7th December 2017 4 / 16

Convex Projective Structures

Convex Projective Structures

A convex projective structure of a surface S is a pair (Ω,Γ):i) Ω ⊂ RP2 properly convex domainii) Γ ≤ PGL3(R) discrete subgroupiii) Ω/Γ ∼= S

Alex Casella On Convex Projective Structures: Part I 7th December 2017 4 / 16

Convex Projective Structures

Convex Projective Structures

A convex projective structure of a surface S is a pair (Ω,Γ):i) Ω ⊂ RP2 properly convex domainii) Γ ≤ PGL3(R) discrete subgroupiii) Ω/Γ ∼= S

Alex Casella On Convex Projective Structures: Part I 7th December 2017 4 / 16

Convex Projective Structures

Convex Projective Structures

A convex projective structure of a surface S is a pair (Ω,Γ):i) Ω ⊂ RP2 properly convex domainii) Γ ≤ PGL3(R) discrete subgroupiii) Ω/Γ ∼= S

Alex Casella On Convex Projective Structures: Part I 7th December 2017 4 / 16

Convex Projective Structures

Convex Projective Structures

A convex projective structure of a surface S is a pair (Ω,Γ):i) Ω ⊂ RP2 properly convex domainii) Γ ≤ PGL3(R) discrete subgroupiii) Ω/Γ ∼= S

Alex Casella On Convex Projective Structures: Part I 7th December 2017 4 / 16

Convex Projective Structures

Convex Projective Structures

A convex projective structure of a surface S is a pair (Ω,Γ):i) Ω ⊂ RP2 properly convex domainii) Γ ≤ PGL3(R) discrete subgroupiii) Ω/Γ ∼= S

Alex Casella On Convex Projective Structures: Part I 7th December 2017 4 / 16

Convex Projective Structures

Marked Convex Projective Structures

A marked convex projective structure of a surface S is a triple(Ω,Γ, f ):

i) Ω ⊂ RP2 properly convex domain

ii) Γ ≤ PGL3(R) discrete subgroup

iii) f is a chosen homeomorphism f : S→ Ω/Γ

Alex Casella On Convex Projective Structures: Part I 7th December 2017 5 / 16

Developing Map and Holonomy Representation

Developing Map

(S, π) is a universal cover of Sq is the quotient map

Alex Casella On Convex Projective Structures: Part I 7th December 2017 6 / 16

Developing Map and Holonomy Representation

Developing Map

(S, π) is a universal cover of Sq is the quotient map

Alex Casella On Convex Projective Structures: Part I 7th December 2017 6 / 16

Developing Map and Holonomy Representation

Developing Map

(S, π) is a universal cover of Sq is the quotient map

dev : S→ Ω,

Alex Casella On Convex Projective Structures: Part I 7th December 2017 6 / 16

Developing Map and Holonomy Representation

Developing Map

(S, π) is a universal cover of Sq is the quotient map

dev : S→ Ω

P 7→ Q

Alex Casella On Convex Projective Structures: Part I 7th December 2017 6 / 16

Developing Map and Holonomy Representation

Developing Map

(S, π) is a universal cover of Sq is the quotient map

dev : S→ Ω

P 7→ Q

dev′ : S→ Ω

P→ T ·Q, T ∈ Γ

The developing map iswell-defined up to leftmultiplication by Γ.

Alex Casella On Convex Projective Structures: Part I 7th December 2017 6 / 16

Developing Map and Holonomy Representation

Holonomy Representation

A developing map gives an identification between S and Ω

dev : S→ Ω

Alex Casella On Convex Projective Structures: Part I 7th December 2017 7 / 16

Developing Map and Holonomy Representation

Holonomy Representation

A developing map gives an identification between S and Ω

dev : S→ Ω

P 7→ Qγ · P 7→ Tγ ·Q

hol : π1(S)→ Γ

γ 7→ Tγ

Alex Casella On Convex Projective Structures: Part I 7th December 2017 7 / 16

Developing Map and Holonomy Representation

Holonomy Representation

A developing map gives an identification between S and Ω

dev : S→ Ω

P 7→ Qγ · P 7→ Tγ ·Q

hol : π1(S)→ Γ

γ 7→ Tγ

dev′ : P 7→ T ·Qγ · P 7→ T · Tγ · T−1 ·Q

hol′ : γ 7→ T · Tγ · T−1

Alex Casella On Convex Projective Structures: Part I 7th December 2017 7 / 16

Developing Map and Holonomy Representation

Holonomy Representation

A developing map gives an identification between S and Ω

dev : S→ Ω

P 7→ Qγ · P 7→ Tγ ·Q

hol : π1(S)→ Γ

γ 7→ Tγ

dev′ : P 7→ T ·Qγ · P 7→ T · Tγ · T−1 ·Q

hol′ : γ 7→ T · Tγ · T−1

In conclusion: a (marked) convex projective structure (Ω,Γ, f )always comes equipped with a pair (dev,hol), defined up to theaction of Γ:

T · (dev,hol) = (T · dev,T · hol ·T−1), T ∈ Γ.

Alex Casella On Convex Projective Structures: Part I 7th December 2017 7 / 16

Developing Map and Holonomy Representation

Developing Map and Holonomy Representation

In conclusion: a (marked) convex projective structure (Ω,Γ, f ) always comesequipped with a pair (dev,hol), defined up to the action of Γ:

T · (dev,hol) = (T · dev,T · hol ·T−1), T ∈ Γ.

Vice-versa, given a pair (dev,hol), I can reconstruct the projectivestructure by taking:

Ω = dev(S);

Γ = hol(π1(S));

f is the unique map which makesthe diagram commute:

Alex Casella On Convex Projective Structures: Part I 7th December 2017 8 / 16

Developing Map and Holonomy Representation

Developing Map and Holonomy Representation

In conclusion: a (marked) convex projective structure (Ω,Γ, f ) always comesequipped with a pair (dev,hol), defined up to the action of Γ:

T · (dev,hol) = (T · dev,T · hol ·T−1), T ∈ Γ.

Vice-versa, given a pair (dev,hol), I can reconstruct the projectivestructure by taking:

Ω = dev(S);

Γ = hol(π1(S));

f is the unique map which makesthe diagram commute:

Alex Casella On Convex Projective Structures: Part I 7th December 2017 8 / 16

Moduli Spaces

Equivalent Structures

Given two marked convex projective structures (Ω0,Γ0, f0) and(Ω1,Γ1, f1), we will say that they are equivalent if and only if there isa projective transformation T such that:

T · Ω0 = Ω1,T · Γ0 · T−1 = Γ1,f0 = f1 T∗,

whereT∗ : Ω0/Γ0 → Ω1/Γ1.

Alex Casella On Convex Projective Structures: Part I 7th December 2017 9 / 16

Moduli Spaces

Equivalent Structures

Given two marked convex projective structures (Ω0,Γ0, f0) and(Ω1,Γ1, f1), we will say that they are equivalent if and only if there isa projective transformation T such that:

T · Ω0 = Ω1,T · Γ0 · T−1 = Γ1,f0 is isotopic to f1 T∗,

whereT∗ : Ω0/Γ0 → Ω1/Γ1.

T3(S) is the moduli space of all equivalence classes of markedconvex projective structures.

Alex Casella On Convex Projective Structures: Part I 7th December 2017 9 / 16

Moduli Spaces

Closed Surfaces - Goldman

Given two marked convex projective structures (Ω0,Γ0, f0) and (Ω1,Γ1, f1), we willsay that they are equivalent if and only if there is a projective transformation T suchthat:

T · Ω0 = Ω1,T · hol0 ·T−1 = hol1,f0 is isotopic to f1 · T∗,

whereT∗ : Ω0/Γ0 → Ω1/Γ1.

T3(S) is the moduli space of all equivalence classes of marked convex projectivestructures.

Theorem (Goldman, 1990)

If S is a closed, orientable surface of genus g (with negative Eulercharacteristic), then T3(S) ∼= R16g−16.

Alex Casella On Convex Projective Structures: Part I 7th December 2017 10 / 16

Moduli Spaces

Surfaces with Boundary

C is a boundary component of S and 〈γ〉 = π1(C) ≤ π1(S).

Alex Casella On Convex Projective Structures: Part I 7th December 2017 11 / 16

Moduli Spaces

Surfaces with Boundary

C is a boundary component of S and 〈γ〉 = π1(C) ≤ π1(S).

In general, hol(γ) is conjugate to

λ+ 0 00 λ0 00 0 λ−

Alex Casella On Convex Projective Structures: Part I 7th December 2017 11 / 16

Moduli Spaces

Surfaces with Boundary

C is a boundary component of S and 〈γ〉 = π1(C) ≤ π1(S).

In general, hol(γ) is conjugate to

λ+ 0 00 λ0 00 0 λ−

Alex Casella On Convex Projective Structures: Part I 7th December 2017 11 / 16

Moduli Spaces

Surfaces with Boundary

C is a boundary component of S and 〈γ〉 = π1(C) ≤ π1(S).C is an hyperbolic end

Alex Casella On Convex Projective Structures: Part I 7th December 2017 11 / 16

Moduli Spaces

Surfaces with Boundary

C is a boundary component of S and 〈γ〉 = π1(C) ≤ π1(S).C is an hyperbolic end

or C is a cusp

Alex Casella On Convex Projective Structures: Part I 7th December 2017 11 / 16

Moduli Spaces

Surfaces with Boundary

C is a boundary component of S and 〈γ〉 = π1(C) ≤ π1(S).C is an hyperbolic end

or C is a cusp

Alex Casella On Convex Projective Structures: Part I 7th December 2017 11 / 16

Moduli Spaces

Finite Area Structures - Marquis

We can consider structures with only cusps.

Lemma (Marquis)A (marked) convex projective structure with only cusps has finite area.

Let Ω be a properly convex domain ofRP2. The Hilbert metric dΩ is definedby:

dΩ(a, b) := log(|CR[x, a, b, y]|).

Alex Casella On Convex Projective Structures: Part I 7th December 2017 12 / 16

Moduli Spaces

Finite Area Structures - Marquis

We can consider structures with only cusps.

Lemma (Marquis)A (marked) convex projective structure with only cusps has finite area.

Let Ω be a properly convex domain ofRP2. The Hilbert metric dΩ is definedby:

dΩ(a, b) := log(|CR[x, a, b, y]|).

Alex Casella On Convex Projective Structures: Part I 7th December 2017 12 / 16

Moduli Spaces

Finite Area Structures - MarquisWe can consider structures with only cusps.

Lemma (Marquis)A (marked) convex projective structure with only cusps has finite area.

Let Ω be a properly convex domain ofRP2. The Hilbert metric dΩ is definedby:

dΩ(a, b) := log(|CR[x, a, b, y]|).

The moduli space of finite-area marked convex projective structures isT f

3 (S).

Theorem (Marquis, 2010)If S is an orientable surface with genus g and n punctures (with negativeEuler characteristic and at least one puncture), then T f

3 (S) ∼= R16g−16+6n.Alex Casella On Convex Projective Structures: Part I 7th December 2017 12 / 16

Moduli Spaces

Minimal and Maximal Hyperbolic Ends

For infinite-area structures, we have another way to make the modulispace T3(S) finite dimensional.

Minimal Hyperbolic End Maximal Hyperbolic End

T m3 (S) is the moduli space of marked convex projective structures

with minimal or maximal hyperbolic ends.

Alex Casella On Convex Projective Structures: Part I 7th December 2017 13 / 16

Moduli Spaces

Minimal and Maximal Hyperbolic Ends

For infinite-area structures, we have another way to make the modulispace T3(S) finite dimensional.

Minimal Hyperbolic End Maximal Hyperbolic End

T m3 (S) is the moduli space of marked convex projective structures

with minimal or maximal hyperbolic ends.

Alex Casella On Convex Projective Structures: Part I 7th December 2017 13 / 16

Moduli Spaces

Framed Structures - Fock and Goncharov

To analyse T m3 (S) properly, one needs a framing.

A framing is a choice, at eachhyperbolic end, of a pair (V, η):

an eigenvector V ∈ ∂Ω;a line η through V andanother eigenvector.

The moduli space of framed marked convex projective structures isT +

3 (S).

Theorem (Fock and Goncharov, 2007)

If S is an orientable surface with genus g and n punctures (withnegative Euler characteristic and at least one puncture), thenT +

3 (S) ∼= R16g−16+8n>0 .

Alex Casella On Convex Projective Structures: Part I 7th December 2017 14 / 16

Moduli Spaces

Framed Structures - Fock and Goncharov

To analyse T m3 (S) properly, one needs a framing.

A framing is a choice, at eachhyperbolic end, of a pair (V, η):

an eigenvector V ∈ ∂Ω;a line η through V andanother eigenvector.

The moduli space of framed marked convex projective structures isT +

3 (S).

Theorem (Fock and Goncharov, 2007)

If S is an orientable surface with genus g and n punctures (withnegative Euler characteristic and at least one puncture), thenT +

3 (S) ∼= R16g−16+8n>0 .

Alex Casella On Convex Projective Structures: Part I 7th December 2017 14 / 16

Conclusion

Conclusion

Alex Casella On Convex Projective Structures: Part I 7th December 2017 15 / 16

Conclusion

Thank You!

Thank you very much for your attention!

Alex Casella On Convex Projective Structures: Part I 7th December 2017 16 / 16