Finite Covering Spaces of 3-manifolds

Lackenby, Marc

doi:10.1142/9789814324359_0086

Cited by 12 publications

(15 citation statements)

References 60 publications

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“…It follows from Lemma 3.3 that the limit on the right is zero, proving (3)(4)(5)(6)(7)(8)(9)(10)(11)(12). By choosing k N big enough we can thus suppose that…”

Section: Theorem 37mentioning

confidence: 95%

“…We use this and an argument similar to that used to prove Lemma 3.5 to deduce (3)(4)(5)(6)(7)(8)(9)(10)(11).…”

Section: Theorem 37mentioning

confidence: 99%

“…See for example Lackenby [10] for recent progress on this and its links with other conjectures in 3-manifold topology.…”

Section: A Presentation For the Alexander Modulementioning

confidence: 99%

“…In the finite dimensional case, if we have a finite index submodule V 0 of a module V then for any metric on M the equality vol.V / D OEV W V 0 vol.V 0 / holds. Since we are interested in approximation problems the analogue of the index we shall consider for the maximal rank submodule L M is 0 .M=L/ (this is justified by (3)(4)(5)(6)(7)(8)(9)(10) Proof Let L 1 ; L 2 be two free submodules of maximal rank in M . Then L 1 \ L 2 is a submodule of maximal rank, and thus contains a free submodule of maximal rank.…”

Section: B1 Definitionmentioning

confidence: 99%

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Exponential growth of torsion in abelian coverings

Raimbault

2012

Algebr. Geom. Topol.

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We show exponential growth of torsion numbers for links whose first nonzero Alexander polynomial has positive logarithmic Mahler measure. This extends a theorem of Silver and Williams to the case of a null first Alexander polynomial and provides a partial solution for a conjecture of theirs. 57M10; 57M25, 57Q10 IntroductionLet M be a compact three-manifold; the homology groups H i .M / can be written as the direct sums H i .M / tors˚Hi .M / free of a finite abelian group with a finite-rank free abelian group. The torsion summand is nontrivial only for i D 1: H 0 and H 3 are Z or 0, and the universal coefficients theorem for cohomology implies that H 1 .M; @M / is free, and by Poincaré duality it follows that H 2 .M / is also torsion-free. On the other hand, the torsion in H 1 can be arbitrarily large (for example, for lens spaces; see below for hyperbolic examples) and it is believed that "most" 3-manifolds should have a rather large torsion. For example E Kowalski shows in [9, Proposition 7.19] that the first homology group of a "Dunfield-Thurston random 3-manifold" typically has a large torsion subgroup. This paper is concerned with the growth rate of the order of H 1 .M N / tors in a sequence of finite coverings M N of a manifold M . The least precise question that can be asked is whether it is exponential in the degree or not, that is, whether the sequencehas a positive limit (or limit superior). This shall be partially answered here in the case where the M N are abelian coverings converging to a free abelian covering of M . The main motivation to study this question was to provide a partial result towards a conjecture of Silver and Williams on the growth rate of torsion numbers of abelian coverings of complements of links (see Conjecture 6.1] or (0-1) below).Historically, the first context where the growth of torsion in the homology of coverings has been studied is that of cyclic coverings of a knot complement. Let K be an open knotted solid torus in the three-sphere and M D S 3 K ; then M is a compact threemanifold with H 1 .M / D Z. Thus we can consider the infinite cyclic covering M of M and its finite quotients M N , which are the finite coverings of M corresponding to

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“…It follows from Lemma 3.3 that the limit on the right is zero, proving (3)(4)(5)(6)(7)(8)(9)(10)(11)(12). By choosing k N big enough we can thus suppose that…”

Section: Theorem 37mentioning

confidence: 95%

“…We use this and an argument similar to that used to prove Lemma 3.5 to deduce (3)(4)(5)(6)(7)(8)(9)(10)(11).…”

Section: Theorem 37mentioning

confidence: 99%

“…See for example Lackenby [10] for recent progress on this and its links with other conjectures in 3-manifold topology.…”

Section: A Presentation For the Alexander Modulementioning

confidence: 99%

Section: B1 Definitionmentioning

confidence: 99%

See 2 more Smart Citations

Exponential growth of torsion in abelian coverings

Raimbault

2012

Algebr. Geom. Topol.

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“…By Strong Approximation Theorem (see Theorem 3.2. of [9]), this map is a surjection for almost all primes p. Let B denote the Borel subgroup of upper triangular elements in PSL 2 (F). The elements in Γ which land in B under φ form a subgroup that we shall denote with Γ 0 (p).…”

Section: Projective Coversmentioning

confidence: 99%

On the Torsion Homology of Non-Arithmetic Hyperbolic Tetrahedral Groups

Sengun

2012

Int. J. Number Theory

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Coarse non-amenability and covers with small eigenvalues

Arzhantseva

Guentner

2011

Math. Ann.

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Given a closed Riemannian manifold M and a (virtual) epimorphism π 1 (M) ։ F 2 of the fundamental group onto a free group of rank 2, we construct a tower of finite sheeted regular covers {M n } ∞ n=0 of M such that λ 1 (M n ) → 0 as n → ∞. This is the first example of such a tower which is not obtainable up to uniform quasi-isometry (or even up to uniform coarse equivalence) by the previously known methods where π 1 (M) is supposed to surject onto an amenable group.

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Finite Covering Spaces of 3-manifolds

Cited by 12 publications

References 60 publications

Exponential growth of torsion in abelian coverings

Exponential growth of torsion in abelian coverings

On the Torsion Homology of Non-Arithmetic Hyperbolic Tetrahedral Groups

Coarse non-amenability and covers with small eigenvalues

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