Exotic components in linear slices of quasi-Fuchsian groups Yuichi - PowerPoint PPT Presentation

Exotic components in linear slices of quasi-Fuchsian groups Yuichi Kabaya Kyoto University Nara, October 29 2015 1 / 25

Outline S : ori. surface with χ ( S ) < 0 X ( S ) = { ρ : π 1 ( S ) → PSL 2 C } / { ∼ conjugation } (character variety) ∪ AH ( S ) = { [ ρ ] ∈ X ( S ) | ρ : faithful, discrete } By the celebrated Ending Lamination Theorem, AH ( S ) is completely classified ( ∃ explicit parametrization). But the shape of AH ( S ) in X ( S ) is complicated (cf. bumping phenomena, non-local connectivity). AH ( S ) (shaded) in some slice of X ( S ) 2 / 25

Outline AH ( S ) (shaded) in some slice of X ( S ) Aim of this talk Try to understand the shape AH ( S ) in X ( S ) by taking slices. In particular, in terms of exotic projective structures. 3 / 25

Quick overview of Kleinian surface groups H 3 : 3-dim hyperbolic space PSL 2 C is isomorphic to the ori. pres. isometry group of H 3 . For a surface S with χ ( S ) < 0, let X ( S ) = { ρ : π 1 ( S ) → PSL 2 C } / { ∼ conj. by PSL 2 C } (character variety) ∪ AH ( S ) = { [ ρ ] ∈ X ( S ) | faithful, ρ ( π 1 ( S )) is discrete } (If S has punctures, we assume that reps are ‘type-preserving’.) H 3 / ρ ( π 1 ( S )) is a hyp 3-mfd homotopy equiv. to S . If ρ ∈ AH ( S ), (Moreover, H 3 / ρ ( π 1 ( S )) is homeo to S × ( − 1 , 1) (Bonahon).) ∼ = Simple example : ρ : π 1 ( S ) − → Γ < PSL 2 ( R ) ( Γ : Fuchsian group) 4 / 25

Quick overview of Kleinian surface groups ∼ = Simple example : ρ : π 1 ( S ) − → Γ < PSL 2 ( R ) ( Γ : Fuchsian group) In this case, the limit set Λ = { accumulation pts of ρ ( π 1 ( S )) · p at ∞ } ⊂ C P 1 (for some p ∈ H 3 ) is a round circle. ρ ∈ AH ( S ) is called quasi-Fuchsian if the limit set Λ is homeo to a circle. QF ( S ) = { ρ ∈ AH ( S ) | quasi-Fuchsian } Anyway, known that QF ( S ) = Int( AH ( S )) . Moreover, QF ( S ) = AH ( S ) (Density Theorem). 5 / 25

Quick overview of Kleinian surface groups By Ahlfors-Bers theorem, QF ( S ) ∼ = T ( S ) × T ( S ) where T ( S ) is the Teichm¨ uller space of S . In particular, QF ( S ) is homeo to R 2(6 g − 6) if S is closed, genus g . X ( S ) = { ρ : π 1 ( S ) → PSL 2 C } / { ∼ conj. by PSL 2 C } ∪ AH ( S ) = { [ ρ ] ∈ X ( S ) | faithful, ρ ( π 1 ( S )) is discrete } ∪ open, dense QF ( S ) = { [ ρ ] ∈ AH ( S ) | quasi-Fuchsian } ∼ = T ( S ) × T ( S ) ∼ = R 2(6 g − 6) 6 / 25

Complex projective structures S : surface ( χ ( S ) < 0) 1 Definition in CP S A complex projective structure or C P 1 -structure on S is a geometric structure PSL(2,C) locally modelled on C P 1 with transition functions in PSL 2 C . (If S has punctures, assume some boundary conditions.) By analytic continuation, we have a pair of maps S → C P 1 (developing map) , D : � ρ : π 1 ( S ) → PSL 2 C (holonomy) s.t. D ( γ · x ) = ρ ( γ ) · D ( x ) ( γ ∈ π 1 ( S ), x ∈ � S ). ρ ( γ ) U U γ ∼ γ Conversely, the pair determines the C P 1 -str (mod ( D , ρ ) ∼ ( gD , g ρ g − 1 )). 7 / 25

Complex projective structures Example (Fuchsian uniformization) = H 2 . Since H 2 ⊂ C P 1 , S ∼ A hyperbolic str on S gives an identification � this gives a C P 1 -str. Similarly as Teichm¨ uller space, we can define P ( S ) = { marked C P 1 -structures on S } . Two important maps : The holonomy gives a map hol : P ( S ) → X ( S ) = Hom( π 1 ( S ) , PSL 2 C ) / conj. : ( D , ρ ) �→ ρ obius transformations are holomorphic, a C P 1 -str defines a Since M¨ hol str (and the hyp. str. conformally equiv. to that). P ( S ) → T ( S ) = Teichm¨ uller space 8 / 25

Bers slice Each fiber of P ( S ) → T ( S ) is parametrized by H 0 ( X , K 2 X ) = { hol. quad. di ff erentials } via Schwarzian derivatives. In particular, if S is closed, genus g , dim R P ( S ) = dim R T ( S ) + dim R H 0 ( X , K 2 X ) = 2(6 g − 6) The set of C P 1 -strs with q-F holonomy in H 0 ( X , K 2 X ) is open. 0 ∈ H 0 ( X , K 2 X ) corresponds to the Fuchsian uniformization of X . The comp ∋ 0 parametrizes T ( S ). (This gives T × T ∼ = QF .) Image by Y. Yamashita But there are many other components : exotic components. We are interested in similar phenomena in another slice. 9 / 25

Goldman’s classification Let Q 0 = { C P 1 -strs with q-F holonomy with inj. dev. map } ⊂ P ( S ) . Q 0 is a conn. comp. of hol − 1 ( QF ( S )) = { C P 1 -strs with q-F holonomy } . 2 π -grafting c ⊂ S : a simple closed curve S → C P 1 by inserting C P 1 along For ( D , ρ ) ∈ Q 0 , we can change D : � each lift of c . This dose not change the holonomy ρ . Q c = { 2 π -grafting of ( D , ρ ) ∈ Q 0 } ⊂ P ( S ) ML Z ( S ) = { disjoint union of scc’s with Z ≥ 0 weight } The above operation can be generalized for µ ∈ ML Z ( S ). Theorem (Goldman (1987)) � hol − 1 ( QF ( S )) = (Q 0 : standard, Q µ ( µ ̸ = 0) : exotic) Q µ µ ∈ ML Z ( S ) 10 / 25

More on 2 π -grafting We have defined 2 π -grafting for Q 0 . This gives ∼ = Q 0 − → Q µ We can also define 2 π -grafting for Q α along β ( α , β ∈ ML Z ( S )). But if the intersection number i ( α , β ) ̸ = 0, it depends on the choice of β in its isotopy class. ∼ = Q α − → Q ( α , β ) ♯ or Q ( α , β ) ♭ β α (Kentaro Ito (2007), Calsamiglia-Deroin-Francaviglia (2014)) 11 / 25

Linear slice ρ ( γ ) ∈ PSL 2 C acts on H 3 . For γ ∈ π 1 ( S ) and ρ ∈ X ( S ), Define the complex length X ( S ) → C / 2 π √− 1 Z by √ λ γ ( ρ ) = (translation length of ρ ( γ )) + − 1 (rotation angle of ρ ( γ )) . This is characterized by � λ γ ( ρ ) � tr( ρ ( γ )) = 2 cosh . 2 S From now on, we assume that S is a once punctured torus. α β For convenience, fix α , β ∈ π 1 ( S ) as in the figure. In this case, dim C X ( S ) = 2. For ℓ > 0, define the linear slice by X ( ℓ ) = { ρ ∈ X ( S ) | λ α ( ρ ) ≡ ℓ } Then dim C X ( ℓ ) = 1, so easy to visualize. 12 / 25

Complex Fenchel-Nielsen coordinates The complex Fenchel-Nielsen coordinates give a parametrization ∼ = { τ ∈ C | − π < Im( τ ) ≤ π } − − → X ( ℓ ) QF ( S ) in the linear slice X (18 . 0). Geometrically speaking, if we let τ = t + √− 1 b , the representation is obtained by twisting distance t and bending with angle b along α . β α t b 13 / 25

Linear slices of QF ( S ) For each ℓ > 0, we are interested in the shape of QF ( ℓ ) := QF ( S ) ∩ X ( ℓ ) ⊂ X ( ℓ ) QF (2 . 0) The Dehn twist along α acts on X ( ℓ ) as τ �→ τ + ℓ . (translation) The real line { τ | Im( τ ) = 0 } corresponds to the Fuchsian representations satisfying λ α = ℓ . By McMullen’s disk convexity of QF ( S ), QF ( ℓ ) is a union of (open) disks. 14 / 25

Linear slices of QF ( S ) For each ℓ > 0, we are interested in the shape of QF ( ℓ ) := QF ( S ) ∩ X ( ℓ ) ⊂ X ( ℓ ) QF (6 . 0) The Dehn twist along α acts on X ( ℓ ) as τ �→ τ + ℓ . (translation) The real line { τ | Im( τ ) = 0 } corresponds to the Fuchsian representations satisfying λ α = ℓ . By McMullen’s disk convexity of QF ( S ), QF ( ℓ ) is a union of (open) disks. 14 / 25

Linear slices of QF ( S ) For any ℓ > 0, there exists a unique standard component containing Fuchsian representations. As pictures suggest; Theorem (Komori-Yamashita, 2012) QF ( ℓ ) has only one component if ℓ is su ffi ciently small, has more than one component if ℓ is su ffi ciently large. QF (2 . 0) QF (6 . 0) We will give another proof for the latter part. In fact, we characterize other components in terms of Goldman’s classification. We lift the slice X ( ℓ ) ⊂ X ( S ) to P ( S ) by complex earthquake. 15 / 25

Grafting (Remark : “grafting” here is similar but di ff erent from 2 π -grafting before. In fact, “grafting” here changes the holonomy.) We can construct another C P 1 -str from a Fuchsian uniformization. X : a hyp str on S , α ⊂ X : a simple closed geodesic. Let Gr b · α ( X ) be the C P 1 -str obtained from X by α inserting a height b annulus along α . In the universal cover � X , the local picture looks like: ~ α b (By construction, Gr 2 π · α ( X ) is obtained from X by 2 π -grafting along α .) 16 / 25

Grafting The grafting operation Gr b · α : T ( S ) → P ( S ) can be generalized for measured laminations. Let ML ( S ) be the set of measured laminations. Theorem (Thurston, Kamishima-Tan) Gr : ML ( S ) × T ( S ) → P ( S ) �→ ( µ, X ) Gr µ ( X ) is a homeomorphism (Thurston coordinates). 17 / 25

Complex Earthquake H = { τ = t + √− 1 b ∈ C | b ≥ 0 } . Let Fix ℓ > 0. � � β α Let tw t · α ( X ℓ ) = ∈ T ( S ). t Define Eq : H → P ( S ) by √ Eq( t + − 1 b ) = Gr b · α (tw t · α ( X ℓ )) ∈ P ( S ) By Thurston coordinates, we can regard H ⊂ P ( S ). Simply denote the image of H by Eq( ℓ ). 18 / 25

Complex Earthquake By construction, hol is the natural projection: hol − − → P ( S ) X ( S ) ⊂ ⊂ → Eq( ℓ ) X ( ℓ ) = = { τ | Im( τ ) ≥ 0 } { τ | − π < Im( τ ) ≤ π } ∈ ∈ τ mod 2 π √− 1 τ �→ We are interested in QF ( ℓ ) := QF ( S ) ∩ X ( ℓ ) ⊂ X ( ℓ ) , so consider hol − 1 ( QF ( ℓ )) = hol − 1 ( X ( ℓ ) ∩ QF ( S )) = Eq( ℓ ) ∩ hol − 1 ( QF ( S )) . 19 / 25