On the Fuchsian locus of PSL n ( R )-Hitchin components for a pair - PowerPoint PPT Presentation

On the Fuchsian locus of PSL n ( R )-Hitchin components for a pair of pants Yusuke Inagaki Osaka University Topology and Computer 2017 Oct 20, 2017 Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Introduction 1 The Bonahon-Dreyer’s parametrization 2 Parameterizing the Fuchsian locus 3 Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Introduction. Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Teichm¨ uller components S : a compact connected orientable surface with χ ( S ) < 0. M ( S ): the set of complete finite-volumed Riemannian metrics on S . Diff 0 ( S ): the identity component of the diffeomorphism group of S . T ( S ) = M ( S ) / Diff 0 ( S ): the Teichm¨ uller space for S . The Teichm¨ uller space is identified with a space of representations. T ( S ) = { ρ ∈ Hom ( π 1 ( S ) , PSL 2 ( R )) | ρ is discrete and faithful } / Conj . Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Teichm¨ uller components S : a compact connected orientable surface with χ ( S ) < 0. M ( S ): the set of complete finite-volumed Riemannian metrics on S . Diff 0 ( S ): the identity component of the diffeomorphism group of S . T ( S ) = M ( S ) / Diff 0 ( S ): the Teichm¨ uller space for S . The Teichm¨ uller space is identified with a space of representations. T ( S ) = { ρ ∈ Hom ( π 1 ( S ) , PSL 2 ( R )) | ρ is discrete and faithful } / Conj . Theorem (Goldman ’88) The subset of Hom ( π 1 ( S ) , PSL 2 ( R )) / conj, denoted by Fuch 2 ( S ), which consists of discrete, faithful representations is a connected component. Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Teichm¨ uller components S : a compact connected orientable surface with χ ( S ) < 0. M ( S ): the set of complete finite-volumed Riemannian metrics on S . Diff 0 ( S ): the identity component of the diffeomorphism group of S . T ( S ) = M ( S ) / Diff 0 ( S ): the Teichm¨ uller space for S . The Teichm¨ uller space is identified with a space of representations. T ( S ) = { ρ ∈ Hom ( π 1 ( S ) , PSL 2 ( R )) | ρ is discrete and faithful } / Conj . Theorem (Goldman ’88) The subset of Hom ( π 1 ( S ) , PSL 2 ( R )) / conj, denoted by Fuch 2 ( S ), which consists of discrete, faithful representations is a connected component. The component Fuch 2 ( S ) is called Teichm¨ uller component . A representation ρ : π 1 ( S ) → PSL 2 ( R ) is called a Fuchsian representation if ρ is discrete and faithful. (i.e. [ ρ ] ∈ Fuch 2 ( S )). F 2 ( S ): the set of Fuchsian representations. Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

PSL n ( R )-Hitchin components (1) The PSL n ( R ) -representation variety for π 1 ( S ) is the set of PSL n ( R )-representations of π 1 ( S ) with the compact open topology. R n ( S ) = Hom ( π 1 ( S ) , PSL n ( R )) . PSL n ( R ) ↷ R n ( S ): the conjugate action. The PSL n ( R ) -character variety for π 1 ( S ) is the GIT-quotient space X n ( S ) = R n ( S ) // PSL n ( R ) . Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

PSL n ( R )-Hitchin components (1) The PSL n ( R ) -representation variety for π 1 ( S ) is the set of PSL n ( R )-representations of π 1 ( S ) with the compact open topology. R n ( S ) = Hom ( π 1 ( S ) , PSL n ( R )) . PSL n ( R ) ↷ R n ( S ): the conjugate action. The PSL n ( R ) -character variety for π 1 ( S ) is the GIT-quotient space X n ( S ) = R n ( S ) // PSL n ( R ) . Theorem (Hitchin ’92) Suppose that S is closed. For n ≥ 3 { 3 if n: odd # of components of X n ( S ) = 6 if n: even . Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

PSL n ( R )-Hitchin components (2) ι n : PSL 2 ( R ) → PSL n ( R ): the irreducible representation. ( ι n ) ∗ : X 2 ( S ) → X n ( S ) : ( ι n ) ∗ ([ ρ ]) = [ ι n ◦ ρ ]. Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

PSL n ( R )-Hitchin components (2) ι n : PSL 2 ( R ) → PSL n ( R ): the irreducible representation. ( ι n ) ∗ : X 2 ( S ) → X n ( S ) : ( ι n ) ∗ ([ ρ ]) = [ ι n ◦ ρ ]. Definition The PSL n ( R )-Hitchin component for S , denoted by Hit n ( S ), is the connected component of X n ( S ) containing Fuch n ( S ) = ( ι n ) ∗ ( Fuch 2 ( S )). Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

PSL n ( R )-Hitchin components (2) ι n : PSL 2 ( R ) → PSL n ( R ): the irreducible representation. ( ι n ) ∗ : X 2 ( S ) → X n ( S ) : ( ι n ) ∗ ([ ρ ]) = [ ι n ◦ ρ ]. Definition The PSL n ( R )-Hitchin component for S , denoted by Hit n ( S ), is the connected component of X n ( S ) containing Fuch n ( S ) = ( ι n ) ∗ ( Fuch 2 ( S )). We call ρ ∈ R n ( S ) a Hitchin representation if [ ρ ] ∈ Hit n ( S ). H n ( S ): the set of Hitchin representations. Fuch n ( S ): the Fuchsian locus . ι n ◦ ρ ∈ R n ( S ) ( ρ ∈ F 2 ( S )): an n - Fuchsian representation . F n ( S ): the set of n -Fuchsian representations. Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

The Bonahon-Dreyer’s parametrization L : a geodesic maximal oriented lamination on S with finite leaves. h 1 , · · · , h s : biinfinite leaves in L . g 1 , · · · , g t : closed leaves in L . T 1 , · · · , T u : ideal triangles in S \ L . Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

The Bonahon-Dreyer’s parametrization L : a geodesic maximal oriented lamination on S with finite leaves. h 1 , · · · , h s : biinfinite leaves in L . g 1 , · · · , g t : closed leaves in L . T 1 , · · · , T u : ideal triangles in S \ L . Theorem (Bonahon-Dreyer ’14) There exists an onto-homeomorphism Φ L : Hit n ( S ) → R N Φ L ([ ρ ]) = ( τ ρ T i , v i ) , · · · , σ ρ abc ( � d ( h j ) , · · · , σ ρ e ( g k ) , · · · ) . where τ ρ abc , σ ρ d are the triangle, shearing invariant defined by Bonahon-Dreyer. (We will define later.) Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Main result Goal: To describe Fuch n ( S ) explicitly by using the Bonahon-Dreyer’s parametrization for a pair of pants. Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Main result Goal: To describe Fuch n ( S ) explicitly by using the Bonahon-Dreyer’s parametrization for a pair of pants. P : a pair of pants. L : the geodesic maximal lamination on P in the figure below. ρ n ∈ F n ( P ): any n -Fuchsian representation of π 1 ( P ). Theorem (I.) We can explicitly compute Φ L ([ ρ n ]). Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

The Bonahon-Dreyer’s parametrization. Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Anosov property of Hitchin representations A representation ρ : π 1 ( S ) → PSL n ( R ) is called an Anosov representation if ρ lifts to an SL n ( R )-representation whose flat S × ρ R n satisfies some dynamical property. associate bundle T 1 � Hitchin representations are Anosov. Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Anosov property of Hitchin representations A representation ρ : π 1 ( S ) → PSL n ( R ) is called an Anosov representation if ρ lifts to an SL n ( R )-representation whose flat S × ρ R n satisfies some dynamical property. associate bundle T 1 � Hitchin representations are Anosov. Theorem (Labourie ’06, Fock-Goncharov ’06) Let ρ : π 1 ( S ) → PSL n ( R ) be a Hitchin representation. Then there exists a unique continuous ρ -equivariant map ξ ρ : ∂ ∞ � S → Flag ( R n ) with the hyperconvexity and positivity. We call ξ ρ flag curve . (Anosov map, limit map.) Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Construction of the Bonahon-Dreyer’s parametrization. Theorem (Bonahon-Dreyer ’14) There exists an onto-homeomorphism Φ L : Hit n ( S ) → R N Φ L ([ ρ ]) = ( τ ρ abc ( � T i , v i ) , · · · , σ ρ d ( h j ) , · · · , σ ρ e ( g k ) , · · · ) . where τ ρ abc , σ ρ d are the triangle, shearing invariant defined by Bonahon-Dreyer. (We will define later.) ρ ∈ H n ( S ) 1:1 → ξ ρ → τ ρ pqr , σ ρ − − p . Flag curves are characterized by the invariants τ ρ pqr , σ ρ p . Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Triangle invariant Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32

Construction of the BD coordinate(Triangle invariant) ρ, ξ ρ : a Hitchin representation and its flag curve. T i : an ideal triangle in S \ L . T i : a lifting of T i in � � S . v , v ′ , v ′′ : ideal vertices of � T i . Topology and Computer 2017 Oct 20, 2017 Yusuke Inagaki (Osaka Univ.) Fuchsian locus / 32