The places where pseudo-Anosovs with small dilatation live Eiko Kin - PowerPoint PPT Presentation

The places where pseudo-Anosovs with small dilatation live Eiko Kin Tokyo Institute of Technology (joint work with M. Takasawa and S. Kojima) RIMS seminar 2012.5.29

The places where pseudo-Anosovs with small dilatation live 2/ 30 This talk is based on the following papers: [KT0] E. Kin and M. Takasawa, The boundary of a fibered face of the magic 3 -manifold and the asymptotic behavior of the minimal pseudo-Anosovs dilata- tions. preprint (2012) arXiv:1205.2956 [KKT] E. Kin, S. Kojima and M. Takasawa, Minimal dilatations of pseudo- Anosovs generated by the magic 3 -manifold and their asymptotic behavior. preprint (2011) arXiv:1104.3939 [KT1] E. Kin and M. Takasawa, Pseudo-Anosovs on closed surfaces having small entropy and the Whitehead sister link exterior , to appear in “Jounal of the Mathematical Society of Japan” [KT2] E. Kin and M. Takasawa, Pseudo-Anosov braids with small entropy and the magic 3 -manifold , Communications in Analysis and Geometry 19, volume 4 (2011), 705-758.

The places where pseudo-Anosovs with small dilatation live 3/ 30 Mapping class groups Σ = Σ g,n ; closed orientable surface of genus g by removing n punctures Homeo + ( Σ ) = { f : Σ → Σ : ori. pres. homeo. pres. punctures setwise } Mod( Σ ) = Homeo + ( Σ ) / Homeo 0 ( Σ ) We focus on elements φ ∈ Mod( Σ ), called pseudo-Anosov (pA).

The places where pseudo-Anosovs with small dilatation live 4/ 30 ∃ f ∈ φ Theorem 1 (Thurston) . φ ∈ Mod( Σ ) is pseudo-Anosov ⇐ ⇒ such that f is a pseudo-Anosov homeo. ∃ λ > 1, and A homeomorphism f : Σ → Σ is pseudo-Anosov if ∃ F s , F u ; a pair of transverse measured foliations such that λ F s and f ( F u ) = λ F u . f ( F s ) = 1 The constant λ is called the dilatation of f . F s and F u are called the stable and unstable foliation of f . stable foliation unstable foliation 1 1 λ 1/ λ

The places where pseudo-Anosovs with small dilatation live 5/ 30 Invariants of pA mapping classes Let f ∈ φ be a pseudo-Anosov homeomorphism. Then λ ( f ) does not depend on the choice of a representative. • λ ( φ ) := λ ( f ) > 1; dilatation of φ • ent( φ ) := log λ ( f ); entropy of φ • Ent( φ ) := | χ ( Σ ) | log λ ( f ); normalized entropy of φ = | χ ( Σ ) | ent( φ )

The places where pseudo-Anosovs with small dilatation live 6/ 30 Mapping classes and Fibered 3-manifolds From φ ∈ Mod( Σ ), we obtain the mapping torus T ( φ ) = Σ × [0 , 1] / ( x, 0) ∼ ( f ( x ) , 1) , where f ∈ φ is a representative f a fiber Σ of T ( φ ), and a monodromy f of a fibration 図 1 Theorem 2 (Thurston) . φ ∈ Mod( Σ ) is pA ⇐ ⇒ T ( φ ) is a hyperbolic 3 - manifold with finite volume

The places where pseudo-Anosovs with small dilatation live 7/ 30 Minimal dilatations problem Fix a surface Σ = Σ g,n . Spec ( Σ ) := { λ ( φ ) | pseudo-Anosov φ ∈ Mod( Σ ) } . ⋆ There exists a minimum of Spec ( Σ ) (Ivanov) δ g,n := min { λ | λ ∈ Spec ( Σ g,n ) } Problem 1. Determine the explicit value of δ g,n . Describe pseudo- Anosov elements which achieve δ g,n .

The places where pseudo-Anosovs with small dilatation live 8/ 30 The purpose of this talk... ⋆ find sequences of pseudo-Anosovs with small dialtation ⋆ Our conjecture: they could have the minimal dilatation ⋆ These pseudo-Anosovs are coming from a single 3-manifold.

The places where pseudo-Anosovs with small dilatation live 9/ 30 The purpose of this talk... ⋆ Magic manifold N = S 3 \ (3 chain link) 図 2 3 chain link (left), braided link of a 3-braid (right) • N is a hyperbolic, fibered 3-manifold.

The places where pseudo-Anosovs with small dilatation live 10/ 30 Minimal dilatation δ 0 ,n D n : n -punctured disk Mod( D n )(= Homeo + (D n ) / isotopy rel ∂D point wise) < Mod( Σ 0 ,n +1 ) B n ≃ Mod( D n ) (minimal dilatation of n -braids) δ ( D n ) := min { λ ( φ ) | φ ∈ Mod( D n ) , pseudo-Anosov } Clearly, δ ( D n ) ≥ δ 0 ,n +1 Question 1. What is the value of δ ( D n ) ?

The places where pseudo-Anosovs with small dilatation live 11/ 30 m n 図 3 σ m,n ∈ B m + n +1 Theorem 3 (Hironaka-K (2006)) . • σ m,n is pA ⇐ ⇒ | m − n | ≥ 2 • When ( m, n ) = ( g − 1 , g + 1) , √ g log λ ( σ g − 1 ,g +1 ) < log(2 + 3) √ g log λ ( σ g − 1 ,g +1 ) → log(2 + 3) as g → ∞ Corollary 1 (HK (2006)) . log δ 0 ,n ≍ 1 /n ⋆ σ g − 1 ,g +1 ∈ B 2 g +1 has the smallest known dilatation (true for g = 2 , 3)

The places where pseudo-Anosovs with small dilatation live 12/ 30 For m ≥ 3, 1 ≤ p ≤ m − 1, T m,p := ( σ 2 1 σ 2 σ 3 · · · σ m − 1 ) p σ − 2 m − 1 ∈ B m (e.g, T 6 , 1 = σ 2 1 σ 2 σ 3 σ 4 σ 5 σ − 2 = σ 2 1 σ 2 σ 3 σ 4 σ − 1 5 ) 5 By forgetting the 1st strand of T m,p , we can define T ′ m,p ∈ B m − 1 Theorem 4 (KT2) . Let g ≥ 2 . (1) σ g − 1 ,g +1 is conjugate to T ′ 2 g +2 , 2 (2) S 3 \ � T 2 g +2 , 2 ≃ magic manifold N , where � b denotes the braided link of a braid b

The places where pseudo-Anosovs with small dilatation live 13/ 30 We can prove more (see [KT2]) • T m,p is pseudo-Anosov ⇐ ⇒ gcd( m − 1 , p ) = 1 • If T m,p is pseudo-Anosov, then S 3 \ � T m,p ≃ N Remark 1 (potential candidates with the smallest dilatation (KT2)) . Pseudo-Anosov m -braids with the smallest known dilatation are of the form T m,p or T ′ m +1 ,p . (True for m ≤ 8 .) ⋆ The places where the braids T m,p live?

The places where pseudo-Anosovs with small dilatation live 14/ 30 Thurston norm of hyperbolic 3 -manifolds M Thurston norm ∥ · ∥ : H 2 ( M, ∂M ; R ) → R ; For an integral class a ∈ H 2 ( M, ∂M ; Z ), define ∥ a ∥ = min F {| χ ( F ) |} , where the minimum is taken over all oriented surface F embedded in M , such that a = [ F ] and F has no components of non-negative Euler characteristic. ⋆ The surface F which realizes this minimum is denoted by F a . ⋆ The norm ∥ · ∥ defined on integral classes admits a unique continuous extension ∥ · ∥ : H 2 ( M, ∂M ; R ) → R which is linear on the ray through the origin. ⋆ The unit ball U M w.r.t to ∥ · ∥ is a compact, convex polyhedron.

The places where pseudo-Anosovs with small dilatation live 15/ 30 The places where the braids T m,p live Consider the Thurston norm ∥ · ∥ : H 2 ( N, ∂N ; R ) → R α := [ F α ], β := [ F β ], γ := [ F γ ] ∈ H 2 ( N, ∂N ; Z ) ∥ α ∥ = ∥ β ∥ = ∥ γ ∥ = 1 ∆ β axis (1,1,1) (0,1,0) γ axis (0,0,1) (0,0,1) (-1,0,0) α axis (1,0,0) (0,0,-1) (0,-1,0) (-1,-1,-1) Every top dimensional face ∆ of ∂U N is a fibered face

The places where pseudo-Anosovs with small dilatation live 16/ 30 C ∆ := a cone over ∆ through 0 ∀ a ∈ int ( C ∆ ): integral class, the minimal representative F a (i.e, for any a = [ F a ]) becomes a fiber of a fibration of N Take a particular fibered face ∆ = { ( X, Y, Z ) | X + Y − Z = 1 , X ≥ 0 , Y ≥ 0 , X ≥ Z, Y ≥ Z } . • When gcd( m − 1 , p ) = 1, we can talk about the integral class, say a m,p ∈ H 2 ( N, ∂N ; Z ), associated to the monodromy T m,p

The places where pseudo-Anosovs with small dilatation live 17/ 30 Where do the braids T m,p live? Answer (see [KT2]) • (the projective class) a m,p ∈ ∆ 1 ⊂ ∆, where ∆ 1 = { ( X, Y, 0) ∈ ∆ } (Recall : the braid by forgetting the 1st strand of T 2 g +2 , 2 is conjugate to σ g − 1 ,g +1 ) g →∞ a 2 g +2 , 2 = (1 / 2 , 1 / 2 , 0) ∼ (1 , 1 , 0) lim ⋆ The monodromy associated to (1 , 1 , 0) is a 3-braid with the dilatation √ √ 2 + 3. (Geometric proof of g log λ ( σ g − 1 ,g +1 ) → log(2 + 3) as g → ∞ ) (0,1,0) (1,1,1) 1 (0,0,-1) (1,0,0)

The places where pseudo-Anosovs with small dilatation live 18/ 30 Minimal dilatation δ g,n , g > 1 Theorem 5 (Tsai 2009) . For any fixed g > 1 , log δ g,n ≍ log n . n ⋆ This is in contrast with the cases g = 0 , 1. ∃ c g > 0 such that log n c g n < log δ g,n < c g log n ⇒ 1 c g < n log δ g,n ( ⇐ < c g ) n log n ⋆ What is the value of c g ?

The places where pseudo-Anosovs with small dilatation live 19/ 30 (Examples by Tsai) ∃ { f g,n : Σ g,n → Σ g,n } n ∈ N such that log λ ( f g,n ) ≍ log n Given g ≥ 2, n n log λ ( f g,n ) n log δ g,n lim = 2(2 g + 1). (So lim sup ≤ 2(2 g + 1).) log n log n n →∞ n →∞ See [KT0]

The places where pseudo-Anosovs with small dilatation live 20/ 30 ∃ ∞ ly many g ’s such that if we fix such a g , then Thm A. [KT0] n log δ g,n lim sup ≤ 2 . log n n →∞ ∀ g ≥ 2, ∃ { n i } ∞ Thm B. [KT0] i =0 with n i → ∞ such that n i log δ g,n i lim sup ≤ 2 . log n i i →∞

The places where pseudo-Anosovs with small dilatation live 21/ 30 Sketch of proof of Theorem B (useful formula) Let a = ( x, y, z ) ∈ int ( C ∆ ) be a primitive fibered class. (1) ∥ a ∥ = x + y − z . (2) the number of the boundary components of the mini. representa- tive F a = F ( x,y,z ) is equal to gcd( x, y + z ) + gcd( y, z + x ) + gcd( z, x + y ) . (3) the dilatation λ ( x,y,z ) is the largest real root of f ( x,y,z ) ( t ) = t x + y − z − t x − t y − t x − z − t y − z + 1 .