Distortion of surfaces in graph manifolds

Hruska, G. Christopher; Nguyen, Hoang Thanh

doi:10.2140/agt.2019.19.363

Cited by 2 publications

(19 citation statements)

References 23 publications

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“…Since the number of vertical and geometrically finite pieces of

S

is finite, we let

R

be the maximum of the numbers

R_{B}

chosen above. By a similar argument as in the proof of [, Theorem 6.1], we can find a path

γ

connecting

{\tilde{s}}_{0}

h ({\overset{s}{true}}_{0})

that intersects each plane

{\overset{T}{true}}_{i} = {\overset{M}{true}}_{i - 1} \cap {\overset{M}{true}}_{i}

with

i = 1, 2, \dots, k - 1

exactly at one point

y_{i}

and satisfies

false| γ false| ⩽ K d ({\tilde{s}}_{0}, h ({\overset{s}{true}}_{0}))

where the constant

K

depends only on the metric

d

. Here,

| \cdot |

denotes the length of a path with respect to the metric

d

.…”

Section: Distortion Of Surfaces Is Determined By the Almost Fiber Partmentioning

confidence: 90%

“…Since

\overset{g}{true}

is an embedding, it follows that

\tilde{g} false(ℓ false)

and

\tilde{g} false(ℓ^{'} false)

are disjoint lines in

\overset{T}{true}

. We note that, on the one hand the Hausdorff distance of two sets

ℓ

and

ℓ^{'}

with respect to

d_{\tilde{S}}

‐metric is infinite (this follows from [, Lemma 3.2]). On the other hand, the Hausdorff distance of two sets

\tilde{g} false(ℓ false)

and

\tilde{g} false(ℓ^{'} false)

with respect to

d_{\tilde{M}}

–metric is finite.…”

Section: Distortion Of Surfaces Is Determined By the Almost Fiber Partmentioning

confidence: 94%

“…The proof of the following proposition is essentially the same as Proposition 4.15 in . Proposition For each

γ \subset Φ (S)

as in Definition , we define

{normalΛ}_{γ} = \max \{\prod_{i = j}^{k} ξ_{e_{i}} false| 1 ⩽ j ⩽ k ⩽ n\} .

If the spirality of

S

is trivial, then there exists a positive constant

normalΛ

such that

{normalΛ}_{γ} ⩽ Λ

for all directed 1‐cycle

γ

Φ (S)

.…”

Section: Surfaces In Non‐geometric 3‐manifoldsmentioning

confidence: 97%

“…Since the map

\overset{g}{true}

is an embedding, it follows that

\tilde{g} false(ℓ_{1} false)

and

\tilde{g} false(ℓ_{2} false)

are disjoint lines in the plane

\overset{T}{true}

. If

{\tilde{B}}_{2}

is horizontal, this contradicts [, Lemma 6.3]. If

{\tilde{B}}_{2}

is geometrically infinite, this contradicts Remark .…”

Section: Distortion Of Surfaces Is Determined By the Almost Fiber Partmentioning

confidence: 99%

“…Remark The distortion of a horizontal surface subgroup in a Seifert‐fibered space group is linear (see ) and the distortion of a geometrically infinite surface subgroup in a hyperbolic manifold group is exponential (by Subgroup Tameness Theorem).…”

Section: Surfaces In Non‐geometric 3‐manifoldsmentioning

confidence: 99%

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Distortion of surfaces in 3‐manifolds

Nguyen

2019

Journal of Topology

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show abstract

“…Since the number of vertical and geometrically finite pieces of

S

is finite, we let

R

be the maximum of the numbers

R_{B}

chosen above. By a similar argument as in the proof of [, Theorem 6.1], we can find a path

γ

connecting

{\tilde{s}}_{0}

h ({\overset{s}{true}}_{0})

that intersects each plane

{\overset{T}{true}}_{i} = {\overset{M}{true}}_{i - 1} \cap {\overset{M}{true}}_{i}

with

i = 1, 2, \dots, k - 1

exactly at one point

y_{i}

and satisfies

false| γ false| ⩽ K d ({\tilde{s}}_{0}, h ({\overset{s}{true}}_{0}))

where the constant

K

depends only on the metric

d

. Here,

| \cdot |

denotes the length of a path with respect to the metric

d

.…”

Section: Distortion Of Surfaces Is Determined By the Almost Fiber Partmentioning

confidence: 90%

“…Since

\overset{g}{true}

is an embedding, it follows that

\tilde{g} false(ℓ false)

and

\tilde{g} false(ℓ^{'} false)

are disjoint lines in

\overset{T}{true}

. We note that, on the one hand the Hausdorff distance of two sets

ℓ

and

ℓ^{'}

with respect to

d_{\tilde{S}}

‐metric is infinite (this follows from [, Lemma 3.2]). On the other hand, the Hausdorff distance of two sets

\tilde{g} false(ℓ false)

and

\tilde{g} false(ℓ^{'} false)

with respect to

d_{\tilde{M}}

–metric is finite.…”

Section: Distortion Of Surfaces Is Determined By the Almost Fiber Partmentioning

confidence: 94%

“…The proof of the following proposition is essentially the same as Proposition 4.15 in . Proposition For each

γ \subset Φ (S)

as in Definition , we define

{normalΛ}_{γ} = \max \{\prod_{i = j}^{k} ξ_{e_{i}} false| 1 ⩽ j ⩽ k ⩽ n\} .

If the spirality of

S

is trivial, then there exists a positive constant

normalΛ

such that

{normalΛ}_{γ} ⩽ Λ

for all directed 1‐cycle

γ

Φ (S)

.…”

Section: Surfaces In Non‐geometric 3‐manifoldsmentioning

confidence: 97%

“…Since the map

\overset{g}{true}

is an embedding, it follows that

\tilde{g} false(ℓ_{1} false)

and

\tilde{g} false(ℓ_{2} false)

are disjoint lines in the plane

\overset{T}{true}

. If

{\tilde{B}}_{2}

is horizontal, this contradicts [, Lemma 6.3]. If

{\tilde{B}}_{2}

is geometrically infinite, this contradicts Remark .…”

Section: Distortion Of Surfaces Is Determined By the Almost Fiber Partmentioning

confidence: 99%

Section: Surfaces In Non‐geometric 3‐manifoldsmentioning

confidence: 99%

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Distortion of surfaces in 3‐manifolds

Nguyen

2019

Journal of Topology

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show abstract

Subgroup distortion of 3-manifold groups

Nguyen

Sun

2020

Trans. Amer. Math. Soc.

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In this paper, we compute the subgroup distortion of all finitely generated subgroups of all finitely generated 3 3 -manifold groups, and the subgroup distortion in this case can only be linear, quadratic, exponential and double exponential. It turns out that the subgroup distortion of a subgroup of a 3 3 -manifold group is closely related to the separability of this subgroup.

show abstract

Distortion of surfaces in graph manifolds

Abstract: Let S N be an immersed horizontal surface in a 3dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially distorted when the surface is not virtually embedded.

Cited by 2 publications

References 23 publications

Distortion of surfaces in 3‐manifolds

Distortion of surfaces in 3‐manifolds

Subgroup distortion of 3-manifold groups

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