Presentation length and Simon’s conjecture

Agol, Ian; Liu, Yi

doi:10.1090/s0894-0347-2011-00711-x

Cited by 24 publications

(81 citation statements)

References 28 publications

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“…The following bounded spanning lemma is similar to [, Lemma 3.4]. It supplies as crucial an ingredient for our situation.…”

Section: Domination Onto Nongeometric 3‐manifoldsmentioning

confidence: 83%

“…Similarly,

{scriptD}_{e} false(ϕ false) < C

holds for any edge

e \in Edg (Λ)

. The strategy is similar to [, Section 3]. It is actually possible to bound the distortions using the presentation length of

π_{1} false(M false)

as introduced in [, Definition 3.1].…”

Section: Domination Onto Nongeometric 3‐manifoldsmentioning

confidence: 99%

“…The strategy is similar to [, Section 3]. It is actually possible to bound the distortions using the presentation length of

π_{1} false(M false)

as introduced in [, Definition 3.1]. However, to establish a factorization theorem like [, Theorems 3.2 and 3.6] would seem unnecessarily involved for the goal of the present paper.…”

Section: Domination Onto Nongeometric 3‐manifoldsmentioning

confidence: 99%

“…For each

{double-struckH}^{2} \times {double-struckE}^{1}

‐geometric piece

J_{v}

, there is an induced area measure on

j^{- 1} false(J_{v} false) \subset K

, which pulls back the horizontal area measure on

J_{v}

, or in other words, which pulls back the area measure on the hyperbolic base 2‐orbifold. (Intuitively it measures the area after projecting onto the base 2‐orbifold; see [, Subsection 3.3] for details.) Then we define the effective area of

(K, j)

to be sum of the area measures of

j^{- 1} false(J_{v} false) \subset K

for all

v \in Ver (Λ)

, denoted as

{normalArea}_{ρ}^{monospaceeff} false(K, j false) ⩾ 0

.…”

Section: Domination Onto Nongeometric 3‐manifoldsmentioning

confidence: 99%

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Finiteness of nonzero degree maps between 3‐manifolds

Liu

2020

Journal of Topology

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In this paper, it is shown that every orientable closed 3‐manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible nongeometric orientable closed 3‐manifolds. Moreover, given any nonzero integer, as a mapping degree up to sign, every orientable closed 3‐manifold maps with that degree onto only finitely many homeomorphically distinct orientable closed 3‐manifolds.

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“…The following bounded spanning lemma is similar to [, Lemma 3.4]. It supplies as crucial an ingredient for our situation.…”

Section: Domination Onto Nongeometric 3‐manifoldsmentioning

confidence: 83%

“…Similarly,

{scriptD}_{e} false(ϕ false) < C

holds for any edge

e \in Edg (Λ)

. The strategy is similar to [, Section 3]. It is actually possible to bound the distortions using the presentation length of

π_{1} false(M false)

as introduced in [, Definition 3.1].…”

Section: Domination Onto Nongeometric 3‐manifoldsmentioning

confidence: 99%

“…The strategy is similar to [, Section 3]. It is actually possible to bound the distortions using the presentation length of

π_{1} false(M false)

as introduced in [, Definition 3.1]. However, to establish a factorization theorem like [, Theorems 3.2 and 3.6] would seem unnecessarily involved for the goal of the present paper.…”

Section: Domination Onto Nongeometric 3‐manifoldsmentioning

confidence: 99%

“…For each

{double-struckH}^{2} \times {double-struckE}^{1}

‐geometric piece

J_{v}

, there is an induced area measure on

j^{- 1} false(J_{v} false) \subset K

, which pulls back the horizontal area measure on

J_{v}

(K, j)

to be sum of the area measures of

j^{- 1} false(J_{v} false) \subset K

for all

v \in Ver (Λ)

, denoted as

{normalArea}_{ρ}^{monospaceeff} false(K, j false) ⩾ 0

.…”

Section: Domination Onto Nongeometric 3‐manifoldsmentioning

confidence: 99%

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Finiteness of nonzero degree maps between 3‐manifolds

Liu

2020

Journal of Topology

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“…For aspherical orientable compact 3-manifolds, Dehn extensions have been investigated in [1] from a topological perspective. This kind of construction was introduced earlier to define knot invariants, known as generalized knot groups (see Kelly [9], Lin and Nelson [10] and Wada [17]).…”

Section: Dehn Extensionsmentioning

confidence: 99%

A Jørgensen–Thurston theorem for homomorphisms

Liu

2012

Algebr. Geom. Topol.

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In this note, we provide a description of the structure homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jørgensen-Thurston Theorem in hyperbolic geometry. YI LIUfinitely-generated case can be reduced to the finitely-presented case, (Proposition 4.12).Acknowledgement. The author thanks Ian Agol and Daniel Groves for helpful conversations. PreliminariesIn this section, we recall some notions and results related to the topic of this paper.2.1. The Jørgensen-Thurston Theorem. For convenience of our discussion, we adopt the following notations and state the results in terms of Kleinian groups. Cf. [Th] for facts mentioned in this subsection.Let Γ be a torsion-free Kleinian group of finite covolume, namely, which has a fundamental domain in H 3 of finite volume. Then Γ has at most finitely many of conjugacy-classes parabolic subgroups, represented by subgroups P 1 , • • • , P q , (q ≥ 0), each isomorphic to a rank-2 free abelian group. We often call these chosen subgroups cusp representatives of Γ. By a slope-tupleor ambiguously, w.r.t. Γ), we shall mean that for each 1 ≤ j ≤ q, the j-th component ζ j is either trivial or a primitive element in P j . For any slopetuple ζ of Γ, we denote the Dehn filling of Γ along ζ as:i.e. Γ quotienting out the normal closure of the ζ j 's, and often denote the quotient epimorphism as:By Thurston's Hyperbolic Dehn-Filling Theorem ([Th, Theorem 5.8.2]), for generic slope-tuples ζ (avoiding finitely many primitive choices for each component), Γ ζ is isomorphic to a torision-free Kleinian group of finite covolume, indeed, strictly less than that of Γ if ζ is nonempty. These Γ ζ 's are usually called hyperbolic Dehn fillings of Γ. In fact, one may choose faithful Kleinian representations ρ ζ : Γ ζ → PSL 2 (C), so that for any sequence {ζ n } of slope-tuples, there is a subsequence for which the induced representations {ρ ζn • ι ζn } of Γ strongly converges. Moreover, if for each j-th component, {ζ j n } has no bounded subsequence of primitive elements in P j , then {ρ ζn • ι ζn } strongly converges to the inclusion Γ ⊂ PSL 2 (C).The following rephrases [Th, Theorem 5.12.1]:Theorem 2.1 (Jørgensen-Thurston). For any V > 0, there exist finitely many torsion-free Kleinian groups Γ 1 , • • • , Γ k of covolume at most V , (together with chosen cusp representatives), such that any torsion-free Kleinian group of covolume at most V is isomorphic to the hyperbolic Dehn filling (Γ i ) ζ of some Γ i along some slope-tuple ζ of Γ i .Remark 2.2. It is also implied from the proof that hyperbolic Dehn fillings decreases the covolume.

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