Braids and Nielsen-Thurston types of automorphisms of punctured surfaces

Braids and

Nielsen-Thurston types of automorphisms of

punctured surfaces

市原 一裕

奈良女子大理学部情報科学科

日本学術振興会 特別研究員

(PD)

共同研究者：

茂手木公彦氏（日本大学文理学部）

§ 1. Introduction

f : automorphism of a compact, ori.

surface with negative Euler chara.

Thm.(Nielsen-Thurston types) f is isotopic to be

(1) periodic

i.e. f ⁿ = id. for some n ∈ Z , or (2) reducible

i.e. f (C ) = C for some ess.

1-submfd. C , or (3) pseudo-Anosov.

Remark ^{: f 6∼} =(1) nor (2) ⇒ f =(3). ∼

Setting.

¶ ³

F : compact ori. surface, χ < 0 x ₁ , · · · , x _n : n points on F

D _{x i} : small neighborhood of x _i S (F ; n) :=

{ϕ ∈ Diff(F ) | ϕ satisfies (∗), ϕ = ∼ id.}

(∗)

 



{x ₁ , · · · , x _n } is invariant

D _{x 1} ∪ · · · ∪ D _{x n} is invariant F ˆ := F − int(D _{x 1} ∪ · · · ∪ D _{x n} )

ˆ

ϕ := ϕ| _{F ˆ}

µ ´

- t

t t

· · ·

x

x

x

· · ·

© ϕ © ϕ ˆ

F F ˆ

Question.

For which ϕ ∈ S (F ; n),

is ϕ ˆ isotopic to be pseudo- Anosov?

Aim.

• To find infinitely many pseudo-Anosov maps.

• To see that pseudo-Anosov maps are

‘generic’.

§ 2. Previous result

F : closed, genus≥ 2, ϕ ∈ S (F ; 1), i.e. n = 1

Thm. (Kra) ˆ

ϕ is isotopic to be pseudo-Anosov iff a closed curve associated to ϕ is stably filling.

In our previous preprint;

‘Nielsen-Thurston types of surface-automorphisms

after puncturing surfaces’,

we gave a topological proof of this thm.

Def. (closed curve associated to ϕ)

¶ ³

An oriented closed curve c on F defined by c(t) = J (t, x ₁ ),

where J : I × F → F , isotopy of ϕ to id.

µ ´

x ₀

x

F c

Remark

For ϕ, ϕ ⁰ ∈ S (F ; 1), ϕ ˆ and ϕ ˆ ⁰ are isotopic iff associated curves are homotopic

relative x ₁ . (Birman)

Def. (Stably filling curve)

¶ ³

A closed curve c on F is called sta- bly filling if ∀c ⁰ , freely homotopic to c, intersects every essential s.c.c. on F .

µ ´

F

c

§ 3. Results

Thm 1.

For ϕ ∈ S (F ; n), ˆ

ϕ is isotopic to be pseudo-Anosov

⇐⇒

 



associated braid b ^ϕ

is sufficiently complicated.

(c.f. Imayoshi-Ito-Yamamoto)

For Φ : F × I → F ,

isotopy of ϕ ∈ S (F ; n) to id..

i.e. Φ(x, 0) = ϕ(x) and Φ(x, 1) = x

Def. (associated braid)

¶ ³

b ^ϕ := (t ^ϕ ₁ (I ), . . . , t ^ϕ _n (I ), F × I ), where i-string t ^ϕ _i : I → F × I

is defined by t ^ϕ _i (s) = (Φ(x _i , s), s).

µ ´

Well-definedness of associated braid Setting.

¶ ³

S ( ˆ F ) := {ϕ| _{F ˆ} | ϕ ∈ S (F ; n)}

Br(F ; n): set of braids in F × I

with endpoints {x ₁ , . . . , x _n }×{0, 1}

µ ´

Def.

¶ ³

b, b ⁰ ∈ Br(F ; n)

b = ∼ b ⁰ , equivalent

def .

⇐⇒

 

 

 

 



 

 

 

 

∃amb. isotopy G of F × I s.t.

· G(x) = x for ∀x ∈ F × {0, 1},

· G(F × {t}) = F × {t} for ∀t ∈ I,

· G(b) = b ⁰ .

µ ´

Proposition

Correspondence ϕ 7→ b ^ϕ induces an isomorphim

S ( ˆ F )/isotopy −→ ^{= ∼} Br(F ; n)/equivalence

Def.

¶ ³

b ∈ Br(F ; n) is sufficiently complicated if

(1) b has no parallel families.

(2) b has no peripheral families.

(3) b is stably filling

i.e. For ∀b ⁰ = ∼ b, p(b ⁰ ) is filling,

where p : F × I → F , projection.

µ ´

F {0}

F {1}

Figure 1.1

(1)

h(D I D I)²

peripheral family

(2)

F I

F {1}

F {0}

N

1

2 m

parallel family

Extension of Kra’s thm.

ϕ ∈ S (F ; n)

p : F × I → F , projection

Def.

¶ ³

A system of closed curves associated to ϕ := a system of closed curves on F appearing as p(b ^ϕ ).

µ ´

Let C ^ϕ = {c ^ϕ ₁ , . . . , c ^ϕ _m } be a system of closed curves associated to ϕ.

Remark

Each c ^ϕ _i corresponds to a minimal sub- family {t _{i 1} , . . . , t _{i p} } of b ^ϕ satisfying

p((∪t _{i `</sub> ) ∩ (F × {0})) = p((∪t <sub>i `} ) ∩ (F × {1})).

Thm 2.

If each c ^ϕ _i is primitive and

C ^ϕ is essential and stably filling, ˆ

ϕ is isotopic to be pseudo-Anosov.

Def.

¶ ³

A closed curve on F is primitive

if it cannot be freely homotopic to some power c ^p (p ≥ 2) of a closed curve c on F .

µ ´

Let C = {c ₁ , · · · , c _m }, C ⁰ = {c ⁰ ₁ , · · · , c ⁰ _m }:

two systems of closed curves on F

Def.

¶ ³

(1) C and C ⁰ are equivalent, C ∼ = C ⁰ if c _i = ∼ c ⁰ _i for i = 1, . . . , m.

(2) C : essential

if ∀c _i is essential on F and

c _i 6∼ = c _j , not freely homo. for i 6= j . (3) C : stably filling

if ∀C ⁰ = ∼ C is filling.

µ ´

§ 4. Proof

For ϕ ∈ S (F ; n),

consider the mapping torus via ϕ;

M _ϕ := F × [0, 1]/{(x, 0) = (ϕ(x), 1)}

-

r

r r

· · ·

· · ·

r

r r

· · ·

· · ·

Lemma.

ˆ

ϕ is isotopic to be pseudo-Anosov

iff F × S ¹ − intN (¯ b ^ϕ ) is hyperbolic,

i.e. atoroidal, not Seifert fibered.

Key of Proof

E:=F × S ¹ − intN (¯ b ^ϕ )

We show that E is Seifert fibered iff b ^ϕ is trivial

i.e. b ^ϕ = ({x ∼ ₁ } × I, . . . , {x _n } × I ; F × I ).

‘If’ part is clear: b ^ϕ is trivial

⇒ E = ˆ ∼ F × S ¹ , Seifert fibered.

‘Only if’ part:

Assume: E is Seifert fibered

Claim.

∃Seifert fibration on F × S ¹ s.t.

each compo. of ¯ b ^ϕ is a Seifert fiber.

F has negative Euler chara. ⇒

Seifert fibrations of F × S ¹ are unique.

i.e. ∃h: diffeo. of F × S ¹ , isotopic to id.

s.t. each compo. of ¯ b ^ϕ 7→ {x _i } × S ¹ .

(We need level preserving such diffeo.)

In fact, from h, we can obtain

∃level preserving diffeo. h ⁰ ( = ∼ id.) s.t. h ⁰ | _{F ×{0,1}} = id.,

h ⁰ ({x _i } × I ) = {x _i } × I for each i.

This implies that b ^ϕ is equivalent to the

trivial braid. ¤