Degree-one maps between hyperbolic 3-manifolds with the same volume limit

Soma, Teruhiko

doi:10.1090/s0002-9947-01-02787-8

Cited by 7 publications

(9 citation statements)

References 22 publications

(29 reference statements)

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“…The following lemma is the infinite volume version of Lemma 1 in Soma [So4]. Here the η-inefficiency is the condition with respect to the continuous map ψ.…”

Section: Inefficiency Of Smearing 3-chainsmentioning

confidence: 96%

“…In this section, we recall the notion of simplicial honeycombs which is introduced in [So4] for hyperbolic 3-manifolds of finite volume and show that it is applicable also to hyperbolic 3-manifolds with infinite volume. Similar tools are used also in [So2].…”

Section: Simplicial Honeycombs (Infinite Volume Version)mentioning

confidence: 99%

“…For the proof, we use the fact given in the previous section that most 3-chains in supp(z X (σ)) are η-efficient. Our argument is the infinite volume version of results in [So4] for closed hyperbolic 3-manifolds. In Section 7, by using the property of ψ, we first construct a locally bi-Lipschitz map ϕ (1) : E thick −→ E = ϕ(E) properly homotopic to ϕ| E thick and then extend ϕ (1) to a bi-Lipschitz map Φ E : E −→ E , which proves Theorem A.…”

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confidence: 99%

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Volume and structure of hyperbolic 3-manifolds

Soma

2019

Preprint

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In this paper, we show that Gromov-Thurston's principle works for hyperbolic 3-manifolds of infinite volume and with finitely generated fundamental group. As an application, we have a new proof of Ending Lamination Theorem. Our proof essentially relays only on Maximum Volume Law for hyperbolic 3-simplices.Let f : M −→ M be a proper degree-one map between oriented hyperbolic 3-manifolds of finite volume. In [Th1, Theorem 6.4], Thurston proved by using results of Gromov [Gr] that f is properly homotopic to an isometry if and only if Vol(M ) = Vol(M ). This theorem suggests us Gromov-Thurston's principle on hyperbolic manifolds of dimension three (or more) that "Volume determines the structure". This principle is essentially supported by Maximum Volume Law, which says that a hyperbolic 3-simplex has the maximum volume v 3 = 1.01494 . . . if and only if it is a regular ideal simplex, see [Th1, Chapter 7]. The main tool for connecting the rigidity with the volume is the smearing 3-cycle z M (σ) on M associated with a straight 3-simplex σ : ∆ 3 −→ H 3 , which is introduced in [Th1, Chapter 6]. Now we consider the case when M is an oriented hyperbolic 3-manifold of infinite volume and with finitely generated fundamental group. Then, instead of the volume of M , we use the bounded 3-cocycle ω M on M such that, for any singular 3simplex τ : ∆ 3 −→ M , ω M (τ ) is the oriented volume of the straightened 3-simplex straight(τ ) of τ . Suppose that any ends of M are incompressible and there exists an orientation and parabolic cusp-preserving homeomorphism ϕ : M −→ M to another oriented hyperbolic 3-manifold M . Let Y be any infinite volume submanifold of M , possibly Y = M . Then, for the restriction z Y (σ) of z M (σ) on Y , ω M (z Y (σ)) is infinite. In such a case, we consider an expanding sequence of compact submanifolds X n of Y with ∞ n=1 X n = Y and substitute the restrictions z Xn (σ) for z Y (σ). The map ϕ is said to satisfy a pseudo-inverse inequality of straightened volume (for short an SV-pseudo-inverse inequality) on Y if there exists a constant c 0 > 0 and submanifolds X n as above such thatfor any n and any straight simplex σ : ∆ 3 −→ H 3 with Vol(σ) > 1. The lower bound '1' is chosen just as a constant such that v 3 − 1 is a positive small number.The following theorem is our main result.

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“…The following lemma is the infinite volume version of Lemma 1 in Soma [So4]. Here the η-inefficiency is the condition with respect to the continuous map ψ.…”

Section: Inefficiency Of Smearing 3-chainsmentioning

confidence: 96%

Section: Simplicial Honeycombs (Infinite Volume Version)mentioning

confidence: 99%

mentioning

confidence: 99%

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Volume and structure of hyperbolic 3-manifolds

Soma

2019

Preprint

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“…Many related partial answers to Questions 1 and 2 have already appeared in the literature, see for example [Ro1], [Ro2], [Ro3], [BW], [HWZ1], [HWZ2], [RW], [So1], [So2], [So3], [Re], [WZ], [De], [De1], [Gu], [BCG], [BBW].…”

Section: Introductionmentioning

confidence: 99%

“…There are some partial results for Question 3 in the case of sequences of degree 1 maps (see [Ro1], [So2]), or when the domain and the target have the same simplicial volume (see [So3], [De]). Question 3 is solved in [De] when M is a graph manifold.…”

Section: Introductionmentioning

confidence: 99%

Finiteness of 3-manifolds associated with non-zero degree mappings

Boileau¹,

Rubinstein²,

Wang³

2014

Comment. Math. Helv.

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We prove a finiteness result for the ∂-patterned guts decomposition of all 3-manifolds obtained by splitting a given orientable, irreducible and ∂-irreducible 3-manifold along a closed incompressible surface. Then using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds dominated by a given compact 3-manifold belong, up to homeomorphism, to a finite collection of compact 3-manifolds. We show also that any closed orientable 3-manifold dominates only finitely many integral homology spheres and any compact 3-manifolds orientable 3-manifold dominates only finitely many exterior of knots in S 3 . * Acknowledgements We thank a lot the referee for his carefull reading of our paper, for providing many fine suggestions, and in particular for pointing out the incompleteness of our induction argument in Section 5. We thanks also Y. Liu and H. Sun for their comments on our drafts.

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A Topologised Measure Homology

Berlanga¹

2008

Glasgow Math. J.

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Abstract.A homology theory based on measures, first mentioned by Thurston, is naturally defined here as a functor into the category of locally convex topological vector spaces. It is proved that the first homology space is Hausdorff.2000 Mathematics Subject Classification. 55N35. Introduction.The study of problems involving measure of theoretic and topological properties may demand the construction of algebraic functors on categories that blend measure and topology. This is the case in topological ergodic theory where one encounters various groups of measure preserving homeomorphisms. This present work begins by recalling the definition of a homology functor based on measures and described by Thurston [23,. It was Prof. D. B. A. Epstein who taught me about this theory a long time ago; so I want to express my deepest gratitude to him.The measure homology groups to be defined become locally convex linear spaces under the appropriate weak topology over a suitable category of topological spaces. Two examples, the circle and an Eilenberg-Mac Lane space of type ‫,ޑ(‬ 1), are discussed. The latter is used to prove that the first homology topological vector space is Hausdorff.From the algebraic point of view, the measure homology groups satisfy the Eilenberg-Steenrod axioms for a wide class of topological spaces including metric spaces and are naturally equivalent to the standard singular homology groups with real coefficients for at least the large class of CW-complexes [11]. Zastrow [27] suggests 'slight changes to Milnor's and Thurston's original definition (giving differences for wild topological spaces only)' (see [27,). In addition, the author gives an example of a (wild) space for which the first measure homology group differs from the first real singular homology group.The homology theory considered here has been used mainly for one application in three-manifold theory: the fundamental cycle of a hyperbolic manifold can be more elegantly expressed as a uniform measure in this theory than in the classical way as a finite sum of singular simplices. Therefore, this homology theory can be conveniently used in the proof that the volume of a closed hyperbolic manifold is a topological invariant. A short description of this fact can be also found in Ratcliffe's book [17, Section 11.5]. This proof of the invariance of the hyperbolic volume is considerably more elegant and stronger than in Benedetti's and Petronio's book [1, Chapter C], which uses classical singular homology theory.

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Degree-one maps between hyperbolic 3-manifolds with the same volume limit

Cited by 7 publications

References 22 publications

Volume and structure of hyperbolic 3-manifolds

Volume and structure of hyperbolic 3-manifolds

Finiteness of 3-manifolds associated with non-zero degree mappings

A Topologised Measure Homology

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