Higher Teichm uller theory and geodesic currents Alessandra Iozzi - PowerPoint PPT Presentation

Higher Teichm¨ uller theory and geodesic currents Alessandra Iozzi ETH Z¨ urich, Switzerland Topological and Homological Methods in Group Theory Bielefeld, April 5th, 2018 A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 1 / 21

Overview Ongoing program to extend features of Teichm¨ uller space to more general situations. In this talk: Some aspects of the classical Teichm¨ ulller theory A structure theorem for geodesic currents Higher Teichm¨ ulller theory and applications Joint with M.Burger, A.Parreau and B.Pozzetti (in progress) Why Teichm¨ uller theory : relations with complex analysis, hyperbolic geometry, the theory of discrete groups, algebraic geometry, low-dimensional topology, differerential geometry, Lie group theory, symplectic geometry, dynamical systems, number theory, TQFT, string theory,... A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 2 / 21

The classical case Teichm¨ uller space Σ g closed orientable surface of genus g ≥ 0 (for simplicity for the moment with p = 0 punctures) T g ∼ = { complete hyperbolic metrics } / Diff + 0 (Σ) Characterizations: � � ρ : π 1 (Σ g ) → PSL(2 , R ) 1 T g ∼ / PSL(2 , R ) = discrete and injective 2 One connected component in Hom( π 1 (Σ g ) , PSL(2 , R )); 3 Maximal level set of � � π 1 (Σ g ) , PSL(2 , R ) / PSL(2 , R ) → R , such that eu(Σ g , · ): Hom | eu(Σ g , ρ ) | ≤ | χ (Σ g ) | . A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 3 / 21

The classical case Thurston compactification: what to look for and why � � � � � T g ∼ = R 6 g − 6 MCG (Σ g ) := Aut π 1 (Σ g ) / Inn π 1 (Σ g ) Wanted a compactification Θ( T g ) such that: = S 6 g − 7 ⇒ 1 The boundary ∂ Θ( T g ) = Θ( T g ) � Θ( T g ) ∼ Θ( T g ) ∼ = closed ball in R 6 g − 6 ; 2 The action of MCG (Σ g ) extends continuously to ∂ Θ( T g ). Then (1)+(2) ⇒ MCG (Σ g ) acts continuously on Θ( T g ) � classify mapping classes (Brower fixed point theorem). [If g = 1, MCG (Σ 1 ) ∼ = SL(2 , Z ), T 1 ∼ = H , and ∃ a classification of isometries and their dynamics by looking at the fixed points in H .] A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 4 / 21

The classical case Thurston–Bonahon compactification C =homotopy classes of closed curves in Σ g .If [ ρ ] ∈ T g , define → R ≥ 0 ℓ [ ρ ] : C − [ γ ] �→ ℓ ( ρ ( c )) where ℓ ( ρ ( c )) = hyperbolic length of the unique ρ -geodesic in [ γ ]. Thus can define � � R C Θ: T g → P ≥ 0 � � [ ρ ] �− → ℓ [ ρ ] with properties: Θ is an embedding; Θ( T g ) is compact; (1)+(2) from before; Θ( T g ) and ∂ Θ( T g ) can be described geometrically in terms of geodesic currents , measured laminations and intersection numbers . A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 5 / 21

The classical case Geodesic currents Curr(Σ) Σ oriented surface with a complete finite area hyperbolization, Γ = π 1 (Σ) and G ( � Σ) = the set of geodesics in � Σ = H Definition A geodesic current on Σ is a positive Radon measure on G ( � Σ) that is Γ-invariant. Σ) ∼ = ( ∂ H ) (2) = { pairs of distinct points in ∂ H } . Convenient: Identify G ( � Example c ⊂ Σ closed geodesic, γ ≃ ( γ − , γ + ) ∈ ( ∂ H ) (2) lift of c .If � δ c := δ η ( γ − ,γ + ) , η ∈ Γ / � γ � supp δ c = lifts of c to H . A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 6 / 21

The classical case Geodesic currents Curr(Σ) Example Liouville current L = the unique PSL(2 , R )-invariant measure on ( ∂ H ) (2) . Let ∂ H = R ∪ {∞} , so [ x , y ] is well defined. If a , b , c , d ∈ ∂ H are positively oriented, � � L [ d , a ] × [ b , c ] := ln[ a , b , c , d ] , where [ a , b , c , d ] := ( a − c )( b − d ) ( a − b )( c − d ) > 1 . a b d c A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 7 / 21

The classical case Geodesic currents Curr(Σ) Example Measure geodesic lamination (Λ , m ) Λ ⊂ Σ = closed subset of Σ that is the union of disjoint simple geodesics; m = homotopy invariant transverse measure to Λ. Lift to a Γ-invariant measure geodesic lamination on H . The associated geodesic current is m ([ a , b ] × [ c , d ]) := ˜ m ( σ ) , where σ is a (geodesic) arc crossing precisely once all leaves connecting [ a , b ] to [ c , d ]. A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 8 / 21

The classical case Intersection number of two currents Know: If α, β ∈ C , then i ( α, β ) = inf α ′ ∈ α,β ′ ∈ β | α ′ ∩ β ′ | Want: If µ, ν ∈ Curr(Σ), define i ( µ, ν ) so that i ( δ c , δ c ′ ) = i ( c , c ′ ). Definition ⋔ := { ( g 1 , g 2 ) ∈ ( ∂ H ) (2) × ( ∂ H ) (2) : | g 1 ∩ g 2 | = 1 } on which Let G 2 PSL(2 , R ) acts properly.Then i ( µ, ν ) := ( µ × ν )(Γ \G 2 ⋔ ) Properties 1 If δ c , δ c ′ ∈ Curr(Σ) ⇒ i ( δ c , δ c ′ ) = i ( c , c ′ ) and i ( δ c , δ c ) = 0 if and only if c is simple. 2 i ( L , δ c ) = ℓ ( c ) = hyperbolic length of c . A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 9 / 21

The classical case Thurston–Bonahon compactification Theorem (Bonahon, ’88) There is a continuous embedding P ( R C I : P (Curr(Σ g )) → ≥ 0 ) [ µ ] �→ { c �→ i ( µ, c ) } whose image contains the Thurston compactification � � P (Curr(Σ g )) Θ( T g ) ⊂ I . Moreover ∂ Θ( T g ) corresponds to the geodesic currents coming from measured laminations. A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 10 / 21

A structure theorem for geodesic currents A structure theorem for geodesic currents Want to generalize to higher rank. Few observations: Intersection can be thought of as length, although more general; Geodesic currents can be thought of as some kind of degenerate hyperbolic structure with geodesics of zero length; Given µ ∈ Curr(Σ), geodesics of zero µ -intersection arrange themselves ”nicely” in Σ ([Burger–Pozzetti, ’15] for µ -lengths) A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 11 / 21

A structure theorem for geodesic currents A structure theorem for geodesic currents Definition Let µ ∈ Curr(Σ).A closed geodesic is µ -special if 1 i ( µ, δ c ) = 0; 2 i ( µ, δ c ′ ) > 0 for all closed geodesic c ′ with c ⋔ c ′ . In particular: 1 A closed geodesic is simple 2 Special geodesics are pairwise non-intersecting. Thus if E µ = { special geodesics on Σ } , |E µ | ≤ ∞ and one can decompose � Σ = Σ v , v ∈ V µ where ∂ Σ v ⊂ E µ . A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 12 / 21

A structure theorem for geodesic currents A structure theorem for geodesic currents Theorem (Burger–I.–Parreau–Pozzetti, ’17) Let µ ∈ Curr(Σ) .Then � � µ = µ v + λ c δ c , v ∈ V µ c ∈E µ where µ v is supported on geodesics contained in ˚ Σ v . Moreover either 1 i ( µ, δ c ) = 0 for all c ∈ ˚ Σ v , hence µ v = 0 , or 2 i ( µ, δ c ) > 0 for all c ∈ ˚ Σ v .In this case either: inf c i ( µ, δ c ) = 0 and supp µ is a π 1 (Σ v ) -invariant lamination that is 1 surface filling and compactly supported, or inf c i ( µ, δ c ) > 0 . 2 Syst Σ v ( µ ) := inf c ⊂ Σ v i ( µ, δ c ) . A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 13 / 21

Higher Teichm¨ uller theory Higher Teichm¨ uller theory Consider representations into a ”larger” Lie group. real adjoint Lie groups � Hitchin component e.g. SL( n , R ), Sp(2 n , R ). [Techniques: Higgs bundles, hyperbolic dynamics, harmonic maps, cluster algebras (Hitchin, Labourie, Fock–Goncharov) ] Hermitian Lie groups � maximal representations Examples: SU( p , q ) (orthogonal group of a Hermitian form of signature ( p , q )), Sp(2 n , R ) [Techniques: Bounded cohomology, Higgs bundles, harmonic maps ıa Prada–Gothen, Koziarz–Maubon) ] (Toledo, Hern´ andez, Burger–I.–Wienhard, Bradlow–Garc´ semisimple real algebraic of non-compact type � positively ratioed representations (Martone–Zhang) Examples: maximal representations and Hitchin components G real adjoint & Hermitian ⇒ G = Sp(2 n , R ) and Hom Hitchin ( π 1 (Σ) , Sp(2 n , R )) � Hom max ( π 1 (Σ) , Sp(2 n , R )) A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 14 / 21

Higher Teichm¨ uller theory Maximal representations Remark Margulis’ superrigidity does not hold. Can define the Toledo invariant T(Σ , · ) : Hom ( π 1 (Σ) , PSp(2 n , R )) / PSp(2 n , R ) → R that is uniformly bounded | T(Σ , · ) | ≤ | χ (Σ) | rank G Definition ρ is maximal if T(Σ , · ) achieves the maximum value A. Iozzi (ETH Z¨ urich) Higher Teichm¨ uller & geodesic currents THGT 15 / 21