Geometry And Dynamics Of Surface Group Representations - UMD
Geometry and Dynamics of Surface GroupRepresentationsWilliam M. GoldmanDepartment of Mathematics University of MarylandColloque de mathématiques de MontréalCRM/ISM14 September 2007()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
1Enhancing Topology with Geometry2Representation varieties and character varieties3Symplectic geometry4Real projective structures on surfaces()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Geometry through symmetryIn his 1872 Erlangen Program, Felix Klein proposed that a geometry is thestudy of properties of an abstract space X which are invariant under atransitive group G of transformations of X .()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Putting geometric structure on a topological spaceTopology: Smooth manifold Σ with coordinate patches U α ;Charts — diffeomorphismsψαUα ψα (Uα ) XOn components of Uα Uβ , g G such thatg ψα ψβ .Local (G , X )-geometry independent of patch.(Ehresmann 1936): Geometric manifold M modeled on X .()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Putting geometric structure on a topological spaceTopology: Smooth manifold Σ with coordinate patches U α ;Charts — diffeomorphismsψαUα ψα (Uα ) XOn components of Uα Uβ , g G such thatg ψα ψβ .Local (G , X )-geometry independent of patch.(Ehresmann 1936): Geometric manifold M modeled on X .()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Putting geometric structure on a topological spaceTopology: Smooth manifold Σ with coordinate patches U α ;Charts — diffeomorphismsψαUα ψα (Uα ) XOn components of Uα Uβ , g G such thatg ψα ψβ .Local (G , X )-geometry independent of patch.(Ehresmann 1936): Geometric manifold M modeled on X .()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Putting geometric structure on a topological spaceTopology: Smooth manifold Σ with coordinate patches U α ;Charts — diffeomorphismsψαUα ψα (Uα ) XOn components of Uα Uβ , g G such thatg ψα ψβ .Local (G , X )-geometry independent of patch.(Ehresmann 1936): Geometric manifold M modeled on X .()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Putting geometric structure on a topological spaceTopology: Smooth manifold Σ with coordinate patches U α ;Charts — diffeomorphismsψαUα ψα (Uα ) XOn components of Uα Uβ , g G such thatg ψα ψβ .Local (G , X )-geometry independent of patch.(Ehresmann 1936): Geometric manifold M modeled on X .()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Putting geometric structure on a topological spaceTopology: Smooth manifold Σ with coordinate patches U α ;Charts — diffeomorphismsψαUα ψα (Uα ) XOn components of Uα Uβ , g G such thatg ψα ψβ .Local (G , X )-geometry independent of patch.(Ehresmann 1936): Geometric manifold M modeled on X .()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Classfication of geometric structuresBasic question: Given a topology Σ and a geometry X G /H,determine all possible ways of providing Σ with the local geometry of(X , G ).Example: The 2-sphere does not admit Euclidean-geometry structure:6 metrically accurate world atlas.Example: The 2-torus admits a rich moduli space ofEuclidean-geometry structures.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Classfication of geometric structuresBasic question: Given a topology Σ and a geometry X G /H,determine all possible ways of providing Σ with the local geometry of(X , G ).Example: The 2-sphere does not admit Euclidean-geometry structure:6 metrically accurate world atlas.Example: The 2-torus admits a rich moduli space ofEuclidean-geometry structures.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Classfication of geometric structuresBasic question: Given a topology Σ and a geometry X G /H,determine all possible ways of providing Σ with the local geometry of(X , G ).Example: The 2-sphere does not admit Euclidean-geometry structure:6 metrically accurate world atlas.Example: The 2-torus admits a rich moduli space ofEuclidean-geometry structures.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Classfication of geometric structuresBasic question: Given a topology Σ and a geometry X G /H,determine all possible ways of providing Σ with the local geometry of(X , G ).Example: The 2-sphere does not admit Euclidean-geometry structure:6 metrically accurate world atlas.Example: The 2-torus admits a rich moduli space ofEuclidean-geometry structures.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Quotients of domainsSuppose that Ω X is an open subset invariant under a subgroupΓ G such that:Γ is discrete;Γ acts properly and freely on ΩThen M Ω/Γ is a (G , X )-manifold covered by Ω.Convex RPn -structures: Ω RPn convex domain.For example, the projective geometry of the interior of a quadric Ω ishyperbolic geometry. B y x AThe hyperbolic distance is defined by cross-ratios:d(x, y ) log[A, x, y , B]()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Quotients of domainsSuppose that Ω X is an open subset invariant under a subgroupΓ G such that:Γ is discrete;Γ acts properly and freely on ΩThen M Ω/Γ is a (G , X )-manifold covered by Ω.Convex RPn -structures: Ω RPn convex domain.For example, the projective geometry of the interior of a quadric Ω ishyperbolic geometry. B y x AThe hyperbolic distance is defined by cross-ratios:d(x, y ) log[A, x, y , B]()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Quotients of domainsSuppose that Ω X is an open subset invariant under a subgroupΓ G such that:Γ is discrete;Γ acts properly and freely on ΩThen M Ω/Γ is a (G , X )-manifold covered by Ω.Convex RPn -structures: Ω RPn convex domain.For example, the projective geometry of the interior of a quadric Ω ishyperbolic geometry. B y x AThe hyperbolic distance is defined by cross-ratios:d(x, y ) log[A, x, y , B]()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Quotients of domainsSuppose that Ω X is an open subset invariant under a subgroupΓ G such that:Γ is discrete;Γ acts properly and freely on ΩThen M Ω/Γ is a (G , X )-manifold covered by Ω.Convex RPn -structures: Ω RPn convex domain.For example, the projective geometry of the interior of a quadric Ω ishyperbolic geometry. B y x AThe hyperbolic distance is defined by cross-ratios:d(x, y ) log[A, x, y , B]()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Quotients of domainsSuppose that Ω X is an open subset invariant under a subgroupΓ G such that:Γ is discrete;Γ acts properly and freely on ΩThen M Ω/Γ is a (G , X )-manifold covered by Ω.Convex RPn -structures: Ω RPn convex domain.For example, the projective geometry of the interior of a quadric Ω ishyperbolic geometry. B y x AThe hyperbolic distance is defined by cross-ratios:d(x, y ) log[A, x, y , B]()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Quotients of domainsSuppose that Ω X is an open subset invariant under a subgroupΓ G such that:Γ is discrete;Γ acts properly and freely on ΩThen M Ω/Γ is a (G , X )-manifold covered by Ω.Convex RPn -structures: Ω RPn convex domain.For example, the projective geometry of the interior of a quadric Ω ishyperbolic geometry. B y x AThe hyperbolic distance is defined by cross-ratios:d(x, y ) log[A, x, y , B]()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Quotients of domainsSuppose that Ω X is an open subset invariant under a subgroupΓ G such that:Γ is discrete;Γ acts properly and freely on ΩThen M Ω/Γ is a (G , X )-manifold covered by Ω.Convex RPn -structures: Ω RPn convex domain.For example, the projective geometry of the interior of a quadric Ω ishyperbolic geometry. B y x AThe hyperbolic distance is defined by cross-ratios:d(x, y ) log[A, x, y , B]()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Quotients of domainsSuppose that Ω X is an open subset invariant under a subgroupΓ G such that:Γ is discrete;Γ acts properly and freely on ΩThen M Ω/Γ is a (G , X )-manifold covered by Ω.Convex RPn -structures: Ω RPn convex domain.For example, the projective geometry of the interior of a quadric Ω ishyperbolic geometry. B y x AThe hyperbolic distance is defined by cross-ratios:d(x, y ) log[A, x, y , B]()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: A projective tiling by equilateral 60o -trianglesThis tesselation of the open triangular region is equivalent to the tiling ofthe Euclidean plane by equilateral triangles.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: A projective deformation of a tiling of thehyperbolic plane by (60o ,60o ,45o )-triangles.Both domains are tiled by triangles, invariant under a Coxeter groupΓ(3, 3, 4). The first domain is bounded by a conic and enjoys hyperbolicgeometry. The second domain is bounded by C 1 α -convex curve where0 α 1.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: A hyperbolic structure on a surface of genus twoIdentify sides of an octagon to form a closed genus two surface.b2a2a1a2b2 b1b1a1Realize these identifications isometrically for a regular 45 o -octagon.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: A hyperbolic structure on a surface of genus twoIdentify sides of an octagon to form a closed genus two surface.b2a2a1a2b2 b1b1a1Realize these identifications isometrically for a regular 45 o -octagon.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: A hyperbolic structure on a surface of genus twoIdentify sides of an octagon to form a closed genus two surface.b2a2a1a2b2 b1b1a1Realize these identifications isometrically for a regular 45 o -octagon.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Modeling structures on representations of π1fMarked (G , X )-structure on Σ: diffeomorphism Σ M where M is a(G , X )-manifold.Marked (G , X )-structures (fi , Mi ) are isotopic isomorphismφM1 M2 with φ f1 ' f2 .Define the deformation space(D(G ,X ) (Σ) : )Marked (G , X )-structures on Σ /Isotopyη Diff(Σ) acts on marked (G , X )-structures: (f , M) 7 (f η, M).Mapping class groupMod(Σ) : π0 Diff(Σ)acts on D(G ,X ) (Σ).()Surface group representations Colloque de mathématiques de Montréal CRM/ 29
Modeling structures on representations of π1fMarked (G , X )-structure on Σ: diffeomorphism Σ M where M is a(G , X )-manifold.Marked (G , X )-structures (fi , Mi ) are isotopic isomorphismφM1 M2 with φ f1 ' f2 .Define the deformation space(D(G ,X ) (Σ) : )Marked (G , X )-structures on Σ /Isotopyη Diff(Σ) acts on marked (G , X )-structures: (f , M) 7 (f η, M).Mapping class groupMod(Σ) : π0 Diff(Σ)acts on D(G ,X ) (Σ).()Surface group representations Colloque de mathématiques de Montréal CRM/ 29
Modeling structures on representations of π1fMarked (G , X )-structure on Σ: diffeomorphism Σ M where M is a(G , X )-manifold.Marked (G , X )-structures (fi , Mi ) are isotopic isomorphismφM1 M2 with φ f1 ' f2 .Define the deformation space(D(G ,X ) (Σ) : )Marked (G , X )-structures on Σ /Isotopyη Diff(Σ) acts on marked (G , X )-structures: (f , M) 7 (f η, M).Mapping class groupMod(Σ) : π0 Diff(Σ)acts on D(G ,X ) (Σ).()Surface group representations Colloque de mathématiques de Montréal CRM/ 29
Modeling structures on representations of π1fMarked (G , X )-structure on Σ: diffeomorphism Σ M where M is a(G , X )-manifold.Marked (G , X )-structures (fi , Mi ) are isotopic isomorphismφM1 M2 with φ f1 ' f2 .Define the deformation space(D(G ,X ) (Σ) : )Marked (G , X )-structures on Σ /Isotopyη Diff(Σ) acts on marked (G , X )-structures: (f , M) 7 (f η, M).Mapping class groupMod(Σ) : π0 Diff(Σ)acts on D(G ,X ) (Σ).()Surface group representations Colloque de mathématiques de Montréal CRM/ 29
Modeling structures on representations of π1fMarked (G , X )-structure on Σ: diffeomorphism Σ M where M is a(G , X )-manifold.Marked (G , X )-structures (fi , Mi ) are isotopic isomorphismφM1 M2 with φ f1 ' f2 .Define the deformation space(D(G ,X ) (Σ) : )Marked (G , X )-structures on Σ /Isotopyη Diff(Σ) acts on marked (G , X )-structures: (f , M) 7 (f η, M).Mapping class groupMod(Σ) : π0 Diff(Σ)acts on D(G ,X ) (Σ).()Surface group representations Colloque de mathématiques de Montréal CRM/ 29
Modeling structures on representations of π1fMarked (G , X )-structure on Σ: diffeomorphism Σ M where M is a(G , X )-manifold.Marked (G , X )-structures (fi , Mi ) are isotopic isomorphismφM1 M2 with φ f1 ' f2 .Define the deformation space(D(G ,X ) (Σ) : )Marked (G , X )-structures on Σ /Isotopyη Diff(Σ) acts on marked (G , X )-structures: (f , M) 7 (f η, M).Mapping class groupMod(Σ) : π0 Diff(Σ)acts on D(G ,X ) (Σ).()Surface group representations Colloque de mathématiques de Montréal CRM/ 29
Representation varietiesLet π hX1 , . . . , Xn i be finitely generated and G GL(N, R) a linearalgebraic group.The set Hom(π, G ) of homomorphismsπ Genjoys the natural structure of an affine algebraic variety.Evaluation on the generatorsHom(π, G ) G nρ 7 ρ(X1 ), . . . , ρ(Xn ) embeds Hom(π, G ) onto an algebraic subset of GL(N, R) N .Structure is {X1 , . . . , Xn }-independent.Classical topology (and Zariski topology).()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Representation varietiesLet π hX1 , . . . , Xn i be finitely generated and G GL(N, R) a linearalgebraic group.The set Hom(π, G ) of homomorphismsπ Genjoys the natural structure of an affine algebraic variety.Evaluation on the generatorsHom(π, G ) G nρ 7 ρ(X1 ), . . . , ρ(Xn ) embeds Hom(π, G ) onto an algebraic subset of GL(N, R) N .Structure is {X1 , . . . , Xn }-independent.Classical topology (and Zariski topology).()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Representation varietiesLet π hX1 , . . . , Xn i be finitely generated and G GL(N, R) a linearalgebraic group.The set Hom(π, G ) of homomorphismsπ Genjoys the natural structure of an affine algebraic variety.Evaluation on the generatorsHom(π, G ) G nρ 7 ρ(X1 ), . . . , ρ(Xn ) embeds Hom(π, G ) onto an algebraic subset of GL(N, R) N .Structure is {X1 , . . . , Xn }-independent.Classical topology (and Zariski topology).()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Representation varietiesLet π hX1 , . . . , Xn i be finitely generated and G GL(N, R) a linearalgebraic group.The set Hom(π, G ) of homomorphismsπ Genjoys the natural structure of an affine algebraic variety.Evaluation on the generatorsHom(π, G ) G nρ 7 ρ(X1 ), . . . , ρ(Xn ) embeds Hom(π, G ) onto an algebraic subset of GL(N, R) N .Structure is {X1 , . . . , Xn }-independent.Classical topology (and Zariski topology).()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Representation varietiesLet π hX1 , . . . , Xn i be finitely generated and G GL(N, R) a linearalgebraic group.The set Hom(π, G ) of homomorphismsπ Genjoys the natural structure of an affine algebraic variety.Evaluation on the generatorsHom(π, G ) G nρ 7 ρ(X1 ), . . . , ρ(Xn ) embeds Hom(π, G ) onto an algebraic subset of GL(N, R) N .Structure is {X1 , . . . , Xn }-independent.Classical topology (and Zariski topology).()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Representation varietiesLet π hX1 , . . . , Xn i be finitely generated and G GL(N, R) a linearalgebraic group.The set Hom(π, G ) of homomorphismsπ Genjoys the natural structure of an affine algebraic variety.Evaluation on the generatorsHom(π, G ) G nρ 7 ρ(X1 ), . . . , ρ(Xn ) embeds Hom(π, G ) onto an algebraic subset of GL(N, R) N .Structure is {X1 , . . . , Xn }-independent.Classical topology (and Zariski topology).()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Natural symmetriesHom(π, G ) admits an action of Aut(π) Aut(G ):φ 1ραπ π G Gwhere (φ, α) Aut(π) Aut(G ), ρ Hom(π, G ).Preserves the algebraic structure.The quotientHom(π, G )/G : Hom(π, G )/ {1} Inn(G )under the subgroup {1} Inn(G ) Aut(π) Aut(G )is the space of equivalence classes of flat connections on G -bundlesover any space with fundamental group π.Inherits an action ofOut(π) : Aut(π)/Inn(π).()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Natural symmetriesHom(π, G ) admits an action of Aut(π) Aut(G ):φ 1ραπ π G Gwhere (φ, α) Aut(π) Aut(G ), ρ Hom(π, G ).Preserves the algebraic structure.The quotientHom(π, G )/G : Hom(π, G )/ {1} Inn(G )under the subgroup {1} Inn(G ) Aut(π) Aut(G )is the space of equivalence classes of flat connections on G -bundlesover any space with fundamental group π.Inherits an action ofOut(π) : Aut(π)/Inn(π).()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Natural symmetriesHom(π, G ) admits an action of Aut(π) Aut(G ):φ 1ραπ π G Gwhere (φ, α) Aut(π) Aut(G ), ρ Hom(π, G ).Preserves the algebraic structure.The quotientHom(π, G )/G : Hom(π, G )/ {1} Inn(G )under the subgroup {1} Inn(G ) Aut(π) Aut(G )is the space of equivalence classes of flat connections on G -bundlesover any space with fundamental group π.Inherits an action ofOut(π) : Aut(π)/Inn(π).()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Natural symmetriesHom(π, G ) admits an action of Aut(π) Aut(G ):φ 1ραπ π G Gwhere (φ, α) Aut(π) Aut(G ), ρ Hom(π, G ).Preserves the algebraic structure.The quotientHom(π, G )/G : Hom(π, G )/ {1} Inn(G )under the subgroup {1} Inn(G ) Aut(π) Aut(G )is the space of equivalence classes of flat connections on G -bundlesover any space with fundamental group π.Inherits an action ofOut(π) : Aut(π)/Inn(π).()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Natural symmetriesHom(π, G ) admits an action of Aut(π) Aut(G ):φ 1ραπ π G Gwhere (φ, α) Aut(π) Aut(G ), ρ Hom(π, G ).Preserves the algebraic structure.The quotientHom(π, G )/G : Hom(π, G )/ {1} Inn(G )under the subgroup {1} Inn(G ) Aut(π) Aut(G )is the space of equivalence classes of flat connections on G -bundlesover any space with fundamental group π.Inherits an action ofOut(π) : Aut(π)/Inn(π).()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
HolonomyA marked structure determines a developing map Σ̃ X and aholonomy representation π G .Globalize the coordinate charts and coordinate changes respectively.Well-defined up to transformations in G .Holonomy defines a mappingholD(G ,X ) (Σ) Hom(π, G )/GEquivariant respectingMod(Σ) Out π1 (Σ) (Thurston): The mapping hol is a local homeomorphism.For quotient structures, hol is an embedding.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
HolonomyA marked structure determines a developing map Σ̃ X and aholonomy representation π G .Globalize the coordinate charts and coordinate changes respectively.Well-defined up to transformations in G .Holonomy defines a mappingholD(G ,X ) (Σ) Hom(π, G )/GEquivariant respectingMod(Σ) Out π1 (Σ) (Thurston): The mapping hol is a local homeomorphism.For quotient structures, hol is an embedding.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
HolonomyA marked structure determines a developing map Σ̃ X and aholonomy representation π G .Globalize the coordinate charts and coordinate changes respectively.Well-defined up to transformations in G .Holonomy defines a mappingholD(G ,X ) (Σ) Hom(π, G )/GEquivariant respectingMod(Σ) Out π1 (Σ) (Thurston): The mapping hol is a local homeomorphism.For quotient structures, hol is an embedding.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
HolonomyA marked structure determines a developing map Σ̃ X and aholonomy representation π G .Globalize the coordinate charts and coordinate changes respectively.Well-defined up to transformations in G .Holonomy defines a mappingholD(G ,X ) (Σ) Hom(π, G )/GEquivariant respectingMod(Σ) Out π1 (Σ) (Thurston): The mapping hol is a local homeomorphism.For quotient structures, hol is an embedding.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
HolonomyA marked structure determines a developing map Σ̃ X and aholonomy representation π G .Globalize the coordinate charts and coordinate changes respectively.Well-defined up to transformations in G .Holonomy defines a mappingholD(G ,X ) (Σ) Hom(π, G )/GEquivariant respectingMod(Σ) Out π1 (Σ) (Thurston): The mapping hol is a local homeomorphism.For quotient structures, hol is an embedding.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
HolonomyA marked structure determines a developing map Σ̃ X and aholonomy representation π G .Globalize the coordinate charts and coordinate changes respectively.Well-defined up to transformations in G .Holonomy defines a mappingholD(G ,X ) (Σ) Hom(π, G )/GEquivariant respectingMod(Σ) Out π1 (Σ) (Thurston): The mapping hol is a local homeomorphism.For quotient structures, hol is an embedding.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
HolonomyA marked structure determines a developing map Σ̃ X and aholonomy representation π G .Globalize the coordinate charts and coordinate changes respectively.Well-defined up to transformations in G .Holonomy defines a mappingholD(G ,X ) (Σ) Hom(π, G )/GEquivariant respectingMod(Σ) Out π1 (Σ) (Thurston): The mapping hol is a local homeomorphism.For quotient structures, hol is an embedding.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
HolonomyA marked structure determines a developing map Σ̃ X and aholonomy representation π G .Globalize the coordinate charts and coordinate changes respectively.Well-defined up to transformations in G .Holonomy defines a mappingholD(G ,X ) (Σ) Hom(π, G )/GEquivariant respectingMod(Σ) Out π1 (Σ) (Thurston): The mapping hol is a local homeomorphism.For quotient structures, hol is an embedding.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Examples: Hyperbolic structuresHyperbolic geometry: When X H2 and G Isom(H2 ), thedeformation space D(G ,X ) (Σ) identifies with the Fricke-Teichmüllerspace F(Σ) of Σ.Identifies with Teichmüller space (marked conformal structures viaUniformization Theorem.hol embeds F(Σ) as a connected component of Hom(π, G )/G ).F(Σ) R6g 6 and Mod(Σ) acts properly discretely.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Examples: Hyperbolic structuresHyperbolic geometry: When X H2 and G Isom(H2 ), thedeformation space D(G ,X ) (Σ) identifies with the Fricke-Teichmüllerspace F(Σ) of Σ.Identifies with Teichmüller space (marked conformal structures viaUniformization Theorem.hol embeds F(Σ) as a connected component of Hom(π, G )/G ).F(Σ) R6g 6 and Mod(Σ) acts properly discretely.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Examples: Hyperbolic structuresHyperbolic geometry: When X H2 and G Isom(H2 ), thedeformation space D(G ,X ) (Σ) identifies with the Fricke-Teichmüllerspace F(Σ) of Σ.Identifies with Teichmüller space (marked conformal structures viaUniformization Theorem.hol embeds F(Σ) as a connected component of Hom(π, G )/G ).F(Σ) R6g 6 and Mod(Σ) acts properly discretely.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Examples: Hyperbolic structuresHyperbolic geometry: When X H2 and G Isom(H2 ), thedeformation space D(G ,X ) (Σ) identifies with the Fricke-Teichmüllerspace F(Σ) of Σ.Identifies with Teichmüller space (marked conformal structures viaUniformization Theorem.hol embeds F(Σ) as a connected component of Hom(π, G )/G ).F(Σ) R6g 6 and Mod(Σ) acts properly discretely.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Examples: Hyperbolic structuresHyperbolic geometry: When X H2 and G Isom(H2 ), thedeformation space D(G ,X ) (Σ) identifies with the Fricke-Teichmüllerspace F(Σ) of Σ.Identifies with Teichmüller space (marked conformal structures viaUniformization Theorem.hol embeds F(Σ) as a connected component of Hom(π, G )/G ).F(Σ) R6g 6 and Mod(Σ) acts properly discretely.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: CP1 -structuresWhen X CP1 and G PGL(2, C), Poincaré identified D (G ,X ) (Σ)with an affine bundle over F(Σ) whose fiber over a Riemann surface Ris the vector space H 0 (R, K 2 ) of holomorphic quadratic differentials.Thus D(G ,X ) (Σ) R12g 12 and Mod(Σ) acts properly discretely.D(G ,X ) (Σ) contains the space of quasi-Fuchsian representations.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: CP1 -structuresWhen X CP1 and G PGL(2, C), Poincaré identified D (G ,X ) (Σ)with an affine bundle over F(Σ) whose fiber over a Riemann surface Ris the vector space H 0 (R, K 2 ) of holomorphic quadratic differentials.Thus D(G ,X ) (Σ) R12g 12 and Mod(Σ) acts properly discretely.D(G ,X ) (Σ) contains the space of quasi-Fuchsian representations.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: CP1 -structuresWhen X CP1 and G PGL(2, C), Poincaré identified D (G ,X ) (Σ)with an affine bundle over F(Σ) whose fiber over a Riemann surface Ris the vector space H 0 (R, K 2 ) of holomorphic quadratic differentials.Thus D(G ,X ) (Σ) R12g 12 and Mod(Σ) acts properly discretely.D(G ,X ) (Σ) contains the space of quasi-Fuchsian representations.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: CP1 -structuresWhen X CP1 and G PGL(2, C), Poincaré identified D (G ,X ) (Σ)with an affine bundle over F(Σ) whose fiber over a Riemann surface Ris the vector space H 0 (R, K 2 ) of holomorphic quadratic differentials.Thus D(G ,X ) (Σ) R12g 12 and Mod(Σ) acts properly discretely.D(G ,X ) (Σ) contains the space of quasi-Fuchsian representations.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: RP2 -structuresWhen X RP2 and G PGL(3, R), the deformation spaceD(G ,X ) (Σ) is a vector bundle over F(Σ) whose fiber over a Riemannsurface R is the vector space H 0 (R, K 3 ) of holomorphic cubicdifferentials (Labourie, Loftin)For any R-split semisimple G , Hitchin found a contractiblecomponent containing F(Σ).More recently, Labourie showed all representations in this componentare discrete embeddings and that Mod(Σ) acts properly discretely.Markedly different from CP1 -structures, where the holonomyrepresentations may not be discrete embeddings and hol is verycomplicated.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: RP2 -structuresWhen X RP2 and G PGL(3, R), the deformation spaceD(G ,X ) (Σ) is a vector bundle over F(Σ) whose fiber over a Riemannsurface R is the vector space H 0 (R, K 3 ) of holomorphic cubicdifferentials (Labourie, Loftin)For any R-split semisimple G , Hitchin found a contractiblecomponent containing F(Σ).More recently, Labourie showed all representations in this componentare discrete embeddings and that Mod(Σ) acts properly discretely.Markedly different from CP1 -structures, where the holonomyrepresentations may not be discrete embeddings and hol is verycomplicated.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: RP2 -structuresWhen X RP2 and G PGL(3, R), the deformation spaceD(G ,X ) (Σ) is a vector bundle over F(Σ) whose fiber over a Riemannsurface R is the vector space H 0 (R, K 3 ) of holomorphic cubicdifferentials (Labourie, Loftin)For any R-split semisimple G , Hitchin found a contractiblecomponent containing F(Σ).More recently, Labourie showed all representations in this componentare discrete embeddings and that Mod(Σ) acts properly discretely.Markedly different from CP1 -structures, where the holonomyrepresentations may not be discrete embeddings and hol is verycomplicated.()Surface group representationsColloque de mathématiques de Montréal CRM/ 29
Example: RP2 -structuresWhen X RP2 and G PGL(3, R), the deformation spaceD(G ,X ) (Σ) is a vector bundle over F(Σ) whose fiber over a Riemannsurface R is the vector space H 0 (R, K 3 ) of holomorphic cubicdifferentials (Labourie, Loftin)For any R-split semisimple G
hyperbolic geometry. x y B A The hyperbolic distance is de ned by cross-ratios: d(x;y) log[A;x;y;B] Surface group representations Colloque de math ematiques de Montr eal CRM/ISM 14 September 2007 6 / 29. Quotients of domains Suppose that ˆ X is an open subset invariant under a subgroup ˆ G such that:
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course. Instead, we will develop hyperbolic geometry in a way that emphasises the similar-ities and (more interestingly!) the many differences with Euclidean geometry (that is, the 'real-world' geometry that we are all familiar with). §1.2 Euclidean geometry Euclidean geometry is the study of geometry in the Euclidean plane R2, or more .
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