Pseudo-Anosovs with small entropy and the magic 3-manifold E. Kin ...

Pseudo-Anosovs with small entropy and themagic 3-manifoldE. Kin ∗ M. TakasawaWorkshop Simplicial Complexes Arising in Low-DimensionalTopology2009.07.02-03

Pseudo-Anosovs and the magic manifold (2009.07.2) 2/ 25Mapping class groupsΣ = Σ g,n ; an orientable surface of genus g with n puncturesM(Σ); the mapping class group on Σ• A classification of mapping classes φ ∈ M(Σ) (Thurston)(1) reducible, (2) periodic, (3) pseudo-Anosov (pA)

Pseudo-Anosovs and the magic manifold (2009.07.2) 3/ 25Properties of pseudo-AnosovsFact. (Miller, Thurston)φ ∈ M(Σ) is pA⇐⇒ φ n (c) ≠ c for any essential loop c in Σ and for any n > 0⇐⇒ mapping torus T(φ) = Σ × [0, 1]/ (x,0)∼(Φ(x),1) is hyperbolic

Pseudo-Anosovs and the magic manifold (2009.07.2) 4/ 25• Fix a hyperbolic metric m on Σ.Fact. Let φ ∈ M(Σ) be pseudo-Anosov.for any essential simple loop c in Σ,∃1 λ = λ(φ) > 1 such thatwhere l m (·) is the length of (·).λ = limn→∞ l m(φ n (c)) 1/n ,• λ(φ) > 1 is called the dilatation of φ.Question. How do you compute λ = λ(φ)?• Compute the transition matrix M (incident matrix) for the train track map ofφ. Then the largest eigenvalue of M equals λ(φ). (Thurston, Penner, Bestvina-Handel....)

Pseudo-Anosovs and the magic manifold (2009.07.2) 5/ 25pseudo-Anosov braids• Σ = D n ; an n-punctured disk.• B n : n-braid group• ∃ Γ : B n → M(D n ) a homomorphism.• b ∈ B n is said to be pseudo-Anosov if Γ(b) is a pA mapping class.• For a braid b, one defines the braided link b.• mapping torus T(Γ(b)) is homeomorphic to S 3 \ b.bb

Pseudo-Anosovs and the magic manifold (2009.07.2) 6/ 25Two invariants of pA1. volume vol(φ) := volume of the mapping torus T(φ)2. dilatation λ(φ) or entropy ent(φ) = log(λ(φ))Fixing a surface Σ,λ(Σ) := min{λ(φ) | φ ∈ M(Σ)},vol(Σ) := min{vol(φ) | φ ∈ M(Σ)}.• It is hard to compute the minimal dilatation and minimal volume.

Pseudo-Anosovs and the magic manifold (2009.07.2) 7/ 25Qeustion. Are there any relation between the two invariants vol(φ) andent(φ)?Yes!Fact. Fix a surface Σ. Then ∃ B = B(Σ) such that ent(φ) ≥ Bvol(φ)for any pA φ ∈ M(Σ).(1) ent(φ) = ||φ|| T .(2) D −1 vol(φ) ≤ ||φ|| W P ≤ D vol(φ), ( ∃ D = D(Σ)) (Brock)(3) C||φ|| T > ||φ|| W P , ( ∃ C = C(Σ)) (Royden)

Magic manifold• n-chain link C nPseudo-Anosovs and the magic manifold (2009.07.2) 8/ 25non-alternating• They are an important examples for the study of exceptional Dehn surgery• M magic := S 3 \ (3-chain link C 3 )(1) smallest known volume among ori. 3-manifolds having 3 cusps.(2) some manifolds having at most 2 cusps with small volume are obtained fromM magic by Dehn fill. (figure 8 knot, whitehead link, whitehead sister link etc)

Families of braidsPseudo-Anosovs and the magic manifold (2009.07.2) 9/ 25• β m,n and ̂β m,nm n m n• β m,n is obtained from ̂β m,n by forgetting two strands• these braids are pseudo-Ansovs for all m, n ≥ 1 (Hironaka-K)

Pseudo-Anosovs and the magic manifold (2009.07.2) 10/ 254-chain link• L c = L ′ means that S 3 \ L = S 3 \ L ′Lemma 1. For each m, n ≥ 1, we have(1) (braided link of β b m,n ) = c C 4(2) (braided link of β m,n ) is obtained from S 3 \ C 4 by a Dehn filling.Proof. See the picture.cc

DilatationPseudo-Anosovs and the magic manifold (2009.07.2) 11/ 25Proposition 1 (Takasawa-K).lim λ(β m,n) =m,n→∞lim λ(̂β m,n ) = 1m,n→∞

Pseudo-Anosovs and the magic manifold (2009.07.2) 12/ 25• efficient graph map (Bestvina-Handel 1994)• transition matrix M = (m i,j )m i,j = ♯ of times that φ(e i ) (image of edge e i ) crosses e j• λ(β m,n ) = λ( b β m,n ) = the largest eigenvalue of M

Pseudo-Anosovs and the magic manifold (2009.07.2) 13/ 25What’s happened for λ(β m,n ) if m, n go to ∞?You can imagine that dilatation is decreasing as m or n goes to ∞pqpq

• λ(D 3 ) = λ(β 1,1 )Pseudo-Anosovs and the magic manifold (2009.07.2) 14/ 25• λ(D 4 ) = λ(β 2,1 ) = λ(β 1,2 )Question.{β m,n }?No!Can we find braids with the smallest dilatations among

Pseudo-Anosovs and the magic manifold (2009.07.2) 15/ 25A friend σ m,n of β m,nm nProposition 2 (H-K). σ m,n is pseudo-Anosov iff n ≠ m, m + 1. In case σ m,n ispseudo-Anosov, λ(σ m,n ) < λ(β m,n )Key: pA representative of β m,n has a 1-pronged singularity at ∂D.(train track of β m,n )(n+1)-gon(m+1)-gon

Pseudo-Anosovs and the magic manifold (2009.07.2) 16/ 25Bounds on the smallest dilatationLemma 2.(1) λ(σ g−1,g+1 ) = min{λ(σ m,n ) | m + n = 2g}(2) λ(σ g−1,g+1 ) is the largest real root oft 2g+1 − 2t g+1 − 2t g + 1.Theorem 1 (H-K 2006). log λ(D 2g+1 ) < log(2 + √ 3)g

Pseudo-Anosovs and the magic manifold (2009.07.2) 17/ 25Corollary (H-K 2006, Minakawa 2006).log λ(Σ g,0 ) < log(2 + √ 3)g• Recently, Thiffeault and Lanneau determined the minimum value ofthe dillatation of pA mapping classes φ ∈ M(Σ g,0 ) for g = 3, 4, 5 withorientable invariant foliations. Their examples has the smaller dilatationthan H-K’s examples.

Pseudo-Anosovs and the magic manifold (2009.07.2) 18/ 25Magic manifold and small dilatation braidProposition 3 (T-K 2009, Venzke 2008).For each n ≥ 2, S 3 \ σ n−1,n+1 is obtained from M magic by a Dehn filling.Theorem 2 (Farb-Leininger-Margalit 2009). There exists a noncompact,hyperbolic 3-manifold M such that:∃ Dehn fillings on M giving a sequence of hyperbolic fibered manifolds,with n i -punctured disk fibers D ni(n i → ∞), and with monodromyΦ i so that λ(D ni ) = λ(Φ i ).• Thm. 2 does not say which manifold satisfies the property

Pseudo-Anosovs and the magic manifold (2009.07.2) 19/ 25Proposition 3. (restated) For each n ≥ 2, S 3 \ σ n−1,n+1 is obtainedfrom M magic by a Dehn filling.Claim 1. σ −11 σ2 2c= Pretzel(2, 2, −4) c = Pretzel(−2, −2, −2) = C 3 .Cx y x y

Pseudo-Anosovs and the magic manifold (2009.07.2) 20/ 25• σ n−1,n+1 ∈ B 2n+1 and ̂σ = ̂σ n−1,n+1 ∈ B 2n+2n-1 n+1conjugateadd 1 strand2n-2Claim 2. ̂σ n−1,n+1 = σ −11 σ2 2 Θk(n) c= σ−11 σ2 2 , where Θ is the full twistbraid.

Pseudo-Anosovs and the magic manifold (2009.07.2) 21/ 25Question. Are there any braids b other than ̂σ n−1,n+1 such thatS 3 \ b = M magic ?Definition 1 (braid T m,p ).T m,p = (112 · · · m − 1) p−1 112 · · · (m − 2)(m − 1) −1 ∈ B m (i := σ i )• example; T 6,2 and T 5,2• If gcd(m − 1, p) ≠ 1, then T m,p is a reducible braid.• By forgetting the 1st strand, one defines an (m−1)-braid from T ′ m,p.

Pseudo-Anosovs and the magic manifold (2009.07.2) 22/ 25Proposition 4. If gcd(m − 1, p) = 1, then S 3 \ T m,p = M magic .• T 2k,2 ′ ∼ σ k−2,k ∈ B 2k−1 .• There are still ∞ly many braids b other than T m,p such that S 3 \ b = M magic .• We know that dilatations of these braids are not small.

Pseudo-Anosovs and the magic manifold (2009.07.2) 23/ 25For n > 0,M n magic := {φ ∈ M(Σ 0,n ) | T(φ) is homeomorphic to M magic }.Main theorem (T-K). (1) For each n ≥ 4, the minimal dilatation ofmapping classes φ ∈ M n magic is realized by:˛nmapping class2k + 1 (k ≥ 2) T 2k,26 σ 1 σ2σ 2 3 σ 44k + 2 (k ≥ 2) T 4k+1,2k−14 T 3,18k + 4 (k ≥ 1) T 8k+3,2k+18∃ b ∈ B 78k + 8 (k ≥ 1) T 8k+7,2k+1˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛˛(2) The above mapping class realizing the minimal dilatation amongM n magic is unique up to conjugacy.

Pseudo-Anosovs and the magic manifold (2009.07.2) 24/ 25Forgetting braid T ′ m,p of T m,p in Main theorem• T 4,1 ′ = β 1,1 ∈ B 3 has the minimal dilatation (Matsuoka)• T 5,1 ′ = β 2,1 ∈ B 4 has the minimal dilatation (Ko-Los-Song)• T 6,2 ′ ∼ σ 1,3 ∈ B 5 has the minimal dilatation (Ham-Song)• T 2k,2 ′ ∼ σ k−2,k ∈ B 2k−1 (H-K).

Pseudo-Anosovs and the magic manifold (2009.07.2) 25/ 25pseudo-Anosovs on Σ g,0 with small dilatationQuestion. Is there any pseudo-Anosov ψ g ∈ M(Σ g,0 ) with small(est)dilatation such that T(ψ g ) is obtained from M magic by a Dehn filling?Yes!Theorem 3 (T-K). For each g ≥ 2, there exists a Dehn filling of M magicwhich gives a hyperbolic fibered manifold with a closed fiber Σ g,0 of genusg, and with a monodromy Ψ g : Σ g,0 → Σ g,0 such that λ(Ψ g ) is less thanor equal to the largest real root oft 2g+1 − 2t g+1 − 2t g + 1.In particular, in case g = 2, Ψ 2 has the smallest dilatation.