Volumeandcommensurabilityofhyperbolic3-manifolds 双曲 3 次元多様体の通約可能性と体積

双曲 3 次元多様体の通約可能性と体積

Volume and commensurability of hyperbolic 3-manifolds

吉田 はん

（群馬工業高等専門学校）

2016

12

23

目次

1.

2.

3.

4.

1.

Γ ₁

Γ ₂ ⊂ Isom ( H ³ ) are commensurable

⇔ [Γ ₁ : Γ ₁ ∩ Γ ₂ ] < ∞ and [Γ ₂ : Γ ₁ ∩ Γ ₂ ] < ∞

M ₁ = H ^{3 / Γ} 1 and M ₂ = H ^{3 / Γ} 2 are commensurable

⇔ Γ ₁ and Γ ₂ are commensurable up to conjugacy

⇔ M ₁ and M ₂ have a common finite sheeted cover.

“commensurability is an equivalence relation.

今までに知られていること

M ₁ = H ^{3 / Γ} 1

M ₂ = H ^{3 / Γ} 2

commensurable ⇒

・

vol ( M ₁ )

vol ( M ₂ ) ∈ Q

・

Q ^{( tr Γ (2)} ₁ ^{) =} Q ^{( tr Γ (2)} ₂ ) (invariant trace field

)

・

cusp parameter

up to P GL (2 , Q ⁾

・同じ

horoball packing

主結果

M ₁ , M ₂

non-arithmetic ori. cusped hyp. 3-mfd.

0 < | vol ( M ₁ ) − vol ( M ₂ ) | < v ₀

4 = 0 . 25 · · ·

ならば

M ₁

M ₂

incommensurable.

v ₀ = 1 . 0149 · · ·

regular ideal tetrahedron

面角は

^π

3

v ₀ = vol

0 = | vol ( M ₁ ) − vol ( M ₂ ) |

M ₁ , M ₂

commensurable

主結果

M ₁ , M ₂

non-arithmetic ori. cusped hyp. 3-mfd.

0 < | vol ( M ₁ ) − vol ( M ₂ ) | < v ₀

4 = 0 . 25 · · ·

ならば

M ₁

M ₂

incommensurable.

v ₀ = 1 . 0149 · · ·

regular ideal tetrahedron

面角は

^π

3

v ₀ = vol

0 = | vol ( M ₁ ) − vol ( M ₂ ) |

M ₁ , M ₂

commensurable

Def. A Kleinian group Γ is called arithmetic if it is com- mensurable with the group norm 1 elements of an order of quaternion algebra A ramified at all real places over a number field k with exactly one complex place.

Ex: The figure-eight knot complement and the White- head link complement are arithmetic.

Rem

aritmetic hyp. mfds and non-arithmetic hyp. mfds

are incommenurable.

２．準備

[G. Margulis]

non-arithmetic hyp. mfds M ₁ and M ₂ are commen- surable

⇔∃

M ₀ : hyp. mfd(or orbifold) s.t. M ₁

M ₀

finite cover M ₂

M ₀

finite cover

Discrete Subgroups of Semi-simple Lie Groups, Ergeb.

der Math. 17, Springer-Verlag (1989).

次のことが知られている．

Th (Adams, 1990). The six noncompact ori. hyp. 3- orbifolds of volumes less than ^v ₄ ⁰ have volumes ₁₂ ^{v 0} , ^v ₆ ¹ , ^v ₆ ⁰ ,

v ₀

6 , ^5v ₂₄ ⁰ and ^v ₄ ¹ . ( v ₁ = 0 . 915 · · · = is the volume of the regular ideal octahedron/4.)

Th (Neumann-Reid(1990)). These six orbifolds are arith- metic.

Cor. non-arithmetic hyp. 3-orbifold

^v ₄ ⁰

吉田の予想

0 . 343 · · ·

^π

2

v ₁ = vol

4

”Noncompact Hyperbolic 3-Orbifolds of Small Volume”, Topology 90,

次のことが知られている．

Th (Adams, 1990). The six noncompact ori. hyp. 3- orbifolds of volumes less than ^v ₄ ⁰ have volumes ₁₂ ^{v 0} , ^v ₆ ¹ , ^v ₆ ⁰ ,

v ₀

6 , ^5v ₂₄ ⁰ and ^v ₄ ¹ . ( v ₁ = 0 . 915 · · · = volume of the regular ideal octahedron/4.)

Th (Neumann-Reid, 1990). These six orbifolds are arith- metic.

”Notes on Adams small volume orbifolds”, Topology 90 (1990)

Cor. non-arithmetic hyp. 3-orbifold

^{v 0}

4

吉田の予想

0 . 343 · · ·

次のことが知られている．

Th (Adams, 1990). The six noncompact ori. hyp. 3- orbifolds of volumes less than ^v ₄ ⁰ have volumes ₁₂ ^{v 0} , ^v ₆ ¹ , ^v ₆ ⁰ ,

v ₀

6 , ^5v ₂₄ ⁰ and ^v ₄ ¹ . ( v ₁ = 0 . 915 · · · = volume of the regular ideal octahedron/4.)

Th (Neumann-Reid, 1990). These six orbifolds are arith- metic.

”Notes on Adams small volume orbifolds”, Topology 90 (1990)

Cor

non-arithmetic hyp. 3-orbifold

^{v 0}

4

吉田の予想

0 . 343 · · ·

次のことが知られている．

Th (Adams, 1990). The six noncompact ori. hyp. 3- orbifolds of volumes less than ^{v 0}

4 have volumes ^{v 0}

12 , ^{v 1}

6 , ^{v 0}

6 ,

v ₀

6 , ^5v ₂₄ ⁰ and ^v ₄ ¹ . ( v ₁ = 0 . 915 · · · = volume of the regular ideal octahedron/4.)

Th (Neumann-Reid, 1990). These six orbifolds are arith- metic.

Cor 1

non-arithmetic hyp. 3-orbifold

^{v 0}

4

吉田の予想

0 . 343 · · ·

3.

主結果

M ₁ , M ₂

non-arithmetic ori. cusped hyp. 3-mfd.

0 < | vol ( M ₁ ) − vol ( M ₂ ) | < v ₀

4 = 0 . 25 · · ·

ならば

M ₁

M ₂

incommensurable.

v ₀

reg- ular ideal tetrahedron

[

]

M ₁

M ₂

commensurable

non-arithmetic

P _i : M _i → M ₀

n _i -fold covering

となる

M ₀

( i = 1 , 2, n ₁ ̸ = n ₂ )

| vol ( M ₁ ) − vol ( M ₂ ) | = | n ₁ vol ( M ₀ ) − n ₂ vol ( M ₀ ) |

= | n ₁ − n ₂ | vol ( M ₀ )

Cor 1.

non-arithmetic ori. hyp. 3-orbifold

v ₀

4

n ₁ ̸ = n ₂

| vol ( M ₁ ) − vol ( M ₂ ) | ≥ v ₀

4

[

]

M ₁

M ₂

commensurable

non-arithmetic

P _i : M _i → M ₀

n _i -fold covering

となる

M ₀

( i = 1 , 2, n ₁ ̸ = n ₂ )

| vol ( M ₁ ) − vol ( M ₂ ) | = | n ₁ vol ( M ₀ ) − n ₂ vol ( M ₀ ) |

= | n ₁ − n ₂ | vol ( M ₀ )

Cor.1

non-arithmetic ori. hyp. 3-orbifold

^v ₄ ⁰

n ₁ ̸ = n ₂

| vol ( M ₁ ) − vol ( M ₂ ) | ≥ v ₀

4

[

]

M ₁

M ₂

commensurable

non-arithmetic

P _i : M _i → M ₀

n _i -fold covering

となる

M ₀

( i = 1 , 2, n ₁ ̸ = n ₂ )

| vol ( M ₁ ) − vol ( M ₂ ) | = | n ₁ vol ( M ₀ ) − n ₂ vol ( M ₀ ) |

= | n ₁ − n ₂ | vol ( M ₀ )

Cor.1

non-arithmetic ori. hyp. 3-orbifold

^v ₄ ⁰

n ₁ ̸ = n ₂

| vol ( M ₁ ) − vol ( M ₂ ) | ≥ v ₀

4

3.

non-ori.cusped hyp. 3-orbifold

^{v 0}

8

Th. M ₁ , M ₂

non-arithmetic nonorientable hyp 3-orbifolds

0 < | vol ( M ₁ ) − vol ( M ₂ ) | < v ₀

8

M ₁

M ₂

incommensurable.

Marshall

Martin

2012

closed ori. hyp. 3- orbifold

0 . 03905 . . .

π 3

π π 5

3

残りの面角は

^π

2

同様にすると，

Th. M ₁ , M ₂

non-arithmetic ori. closed hyperbolic 3- manifold

0 < | vol ( M ₁ ) − vol ( M ₂ ) | < 0 . 03905 · · ·

M ₁

M ₂

incommensurable.

M : n − cusped hyp. mfd M _(p

1 ,q ₁ , ··· ,p _n ,q _n ) : M

i -cusp

( p _i , q _i ) Dehn filling

hyp. mfd.

とする．

hyperbolic Dehn surgery Theorem

vol ( M _(p

1 ,q ₁ , ··· ,p _n ,q _n ) ) → vol ( M ) ( p ² _i + q _i ² → ∞ )

{ vol ( N ) : N is arith . hyp . mfd }

discrete

結果

M

cusped hyperbolic 3-manifold

{ M

Dehn filling

hyperbolic 3-manifolds }

commensurability

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