Non left-orderable surgeries and generalized Baumslag- Solitar relators

(1)

Non left-orderable surgeries and generalized Baumslag- Solitar relators Y. Temma

Background Left-orderability andL-spaces LO-surgery and L-space surgery

Main Theorem Baumslag- Solitar relator

Non left-orderable surgeries and generalized Baumslag-Solitar

relators

Yuki Temma

Nihon University

College of Humanities and Sciences

Joint work with K. Ichihara (Nihon Univ.)

1 / 12

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..

1 Background

Left-orderability and L-spaces LO-surgery and L-space surgery

..

2 Main Theorem

..

3 Baumslag-Solitar relator

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L-space Conjecture

L-space Conjecture [Boyer-Gordon-Watson, 2011]

M: an irreducible rational homology sphere

M is an L-space if and only if π

1

(M ) is not LO

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Left-orderable and L-space

left-orderability G: a non-trivial group

G is called left-orderable (LO) if there exists a strict total order < on G which is left invariant:

g < h −→ f g < f h for ∀ f ∈ G

L-space

M: a rational homology sphere

HF d (M): Heegaard Floer homology with coeﬃcients in Z

2

M is called an L-space if rk HF d (M) = | H

₁

(M ; Z ) |

For example, lens spaces, more generally, spherical 3-manifolds are L-spaces.

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..

1 Background

Left-orderability and L-spaces LO-surgery and L-space surgery

..

2 Main Theorem

..

3 Baumslag-Solitar relator

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Dehn surgery

A Dehn surgery is one of the simple ways to construct L-spaces.

The following operation to obtain another 3-manifold from a given 3-manifold is called a Dehn surgery.

K: a knot in a 3-manifold M Dehn surgery on K

..

1

remove an open regular neighborhood of K from M (drilling)

..

2

glue a solid torus V back along a slope γ (Dehn filling)

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Surgery slope

K: a knot in the 3-sphere S

³

Notation

For f : ∂V → ∂E(K) and the meridian m of V ,

the slope (i.e., isotopy class) γ of the loop f (m) on ∂E(K) is called the surgery slope.

Such a slope on ∂E(K) can be regarded as r ∈ Q ∪ {1/0}.

Notation

K(r): the manifold obtained by surgery on K along r.

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LO-surgery and L-space surgery

K: a knot in 3-sphere S

³

K(p/q): a 3-manifold obtained by Dehn surgery on K along the slope p/q

left-orderable surgery

A Dehn surgery on K is called a left-orderable surgery if it yields a closed 3-manifold with π

₁

(K(p/q)) is LO.

L-space surgery

A Dehn surgery on K is called an L-space surgery if it yields a closed 3-manifold which is an L-space.

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Question

Which knots have non-LO surgery?

We want to provide a characterization of knots which have non-LO surgery in the future.

Theorem [Lidman-Moore, preprint]

For s ≥ 0, only ( − 2, 3, 2s + 1)-pretzel knots have L-space surgeries among hyperbolic pretzel knots.

Hence, if L-space Conjecture is true, it follows that only ( − 2, 3, 2s + 1)-pretzel knots have non-LO surgeries among hyperbolic pretzel knots.

7 / 12

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..

1 Background

Left-orderability and L-spaces LO-surgery and L-space surgery

..

2 Main Theorem

..

3 Baumslag-Solitar relator

(11)

Background Left-orderability andL-spaces LO-surgery and L-space surgery Main Theorem Baumslag- Solitar relator

Nakae’s result

Theorem [Nakae, 2013]

K

_s

: ( − 2, 3, 2s + 1)-pretzel knot in S

³

(s ≥ 3)

K

s

(p/q): a closed 3-manifold obtained by Dehn surgery on K

s

along a slope p/q

If q > 0, p/q ≥ 4s + 7, then π

1

(K

s

(p/q)) is non left-orderable.

Remark: The following result is showed by Clay-Watson in 2012;

Theorem [Clay-Watson, 2013]

Let K

_s

be a ( − 2, 3, 2s − 1)-pretzel knot. If p/q > 2s + 9 and s ≥ 3, π

1

(K

s

(p/q)) is not left-orderable.

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Main Theorem As an extension of Nakae’s result, we have:

Theorem

K: a knot in a 3-manifold M

Suppose that π

₁

(M − K) has a presentation such as

⟨ a, b | (w

₁

a

^m

w ¯

₁

)b

⁻^r

( ¯ w

₂

a

ⁿ

w

₂

)b

^r⁻^k

⟩

Here w

1

, w

2

are arbitrary words and w ¯

i

denotes the word which satisfies w

_i

w ¯

_i

= 1 for i = 1, 2 with m, n ≥ 0, r ∈ Z , k ≥ 0.

Suppose further that a represents a meridian of K and a

⁻^s

wa

⁻^t

represents a longitude of K with s, t ∈ Z and w a word which excludes a

⁻¹

and b

⁻¹

.

If q ̸ = 0 and p/q ≥ s + t, then Dehn surgery on K along the slope p/q yields a closed 3-manifold with π

₁

(K(p/q)) is non left-orderable.

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Proposition

Suppose that a group G has a presentation such as

⟨a, b | (w

1

a

^m

w ¯

1

)b

⁻^r

( ¯ w

2

a

ⁿ

w

2

)b

^r⁻^k

, M LM

⁻¹

L

⁻¹

, M

^p

L

^q

⟩ Here w

₁

, w

₂

are arbitrary words with m, n ≥ 0, r ∈ Z , k ≥ 0, p, q ∈ Z, M = a, L = a

⁻^s

wa

⁻^t

, w is a word which excludes a

⁻¹

and b

⁻¹

, s, t ∈ Z .

If q ̸ = 0 and p/q ≥ s + t, then every homomorphism Φ : G → Homeo

⁺

( R ) has a global fixed point.

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Proof of Main Theorem

Suppose that π

₁

(K(p/q)) is left-orderable. Then it is known that a countable group G is left-orderable if and only if G is isomorphic with a subgroup of Homeo

⁺

( R ). This implies that there exists an injective homomorphism

Φ : π

1

(K(p/q)) → Homeo

⁺

(R).

It was also proved that if there is a homomorphism

G → Homeo

⁺

( R ) with image ̸ = { id } , then there is another such homomorphism which induces an action on R without global fixed points. Hence we may assume that Φ has no global fixed point from this. This contradicts the previous proposition.

Non left-orderable surgeries and generalized Baumslag- Solitar relators

Non left-orderable surgeries and generalized Baumslag-Solitar

relators

Yuki Temma

Nihon University

College of Humanities and Sciences

Joint work with K. Ichihara (Nihon Univ.)

Table of contents

..

1 Background

Left-orderability and L-spaces LO-surgery and L-space surgery

..

2 Main Theorem

..

3 Baumslag-Solitar relator

L-space Conjecture

L-space Conjecture [Boyer-Gordon-Watson, 2011]

M: an irreducible rational homology sphere

M is an L-space if and only if π

(M ) is not LO

Left-orderable and L-space

left-orderability G: a non-trivial group

G is called left-orderable (LO) if there exists a strict total order < on G which is left invariant:

g < h −→ f g < f h for ∀ f ∈ G

L-space

M: a rational homology sphere

HF d (M): Heegaard Floer homology with coeﬃcients in Z

M is called an L-space if rk HF d (M) = | H

(M ; Z ) |

For example, lens spaces, more generally, spherical 3-manifolds are L-spaces.

Table of contents

..

1 Background

Left-orderability and L-spaces LO-surgery and L-space surgery

..

2 Main Theorem

..

3 Baumslag-Solitar relator

Dehn surgery

A Dehn surgery is one of the simple ways to construct L-spaces.

The following operation to obtain another 3-manifold from a given 3-manifold is called a Dehn surgery.

K: a knot in a 3-manifold M Dehn surgery on K

remove an open regular neighborhood of K from M (drilling)

glue a solid torus V back along a slope γ (Dehn filling)

Surgery slope

K: a knot in the 3-sphere S

Notation

For f : ∂V → ∂E(K) and the meridian m of V ,

the slope (i.e., isotopy class) γ of the loop f (m) on ∂E(K) is called the surgery slope.

Such a slope on ∂E(K) can be regarded as r ∈ Q ∪ {1/0}.

Notation

K(r): the manifold obtained by surgery on K along r.

LO-surgery and L-space surgery

K: a knot in 3-sphere S

K(p/q): a 3-manifold obtained by Dehn surgery on K along the slope p/q

left-orderable surgery

A Dehn surgery on K is called a left-orderable surgery if it yields a closed 3-manifold with π

(K(p/q)) is LO.

L-space surgery

A Dehn surgery on K is called an L-space surgery if it yields a closed 3-manifold which is an L-space.

Question

Question

Which knots have non-LO surgery?

We want to provide a characterization of knots which have non-LO surgery in the future.

Theorem [Lidman-Moore, preprint]

For s ≥ 0, only ( − 2, 3, 2s + 1)-pretzel knots have L-space surgeries among hyperbolic pretzel knots.

Hence, if L-space Conjecture is true, it follows that only ( − 2, 3, 2s + 1)-pretzel knots have non-LO surgeries among hyperbolic pretzel knots.

Table of contents

..

1 Background

Left-orderability and L-spaces LO-surgery and L-space surgery

..

2 Main Theorem

..

3 Baumslag-Solitar relator

Nakae’s result

Theorem [Nakae, 2013]

K

: ( − 2, 3, 2s + 1)-pretzel knot in S

(s ≥ 3)