The asymptotic growth of torsion homology for arithmetic groups

Bergeron, Nicolas; Venkatesh, Akshay

doi:10.1017/s1474748012000667

Cited by 133 publications

(257 citation statements)

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“…G Š Z m , for any subgroup H G we denote by X H the Galois covering of X given by 1 .H / n z X ; its Galois group is G=H . We also denote the free abelian covering of X given by ker.…”

Section: Proof Of the Main Theoremsmentioning

confidence: 99%

“…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 the maps 1 .M / ! Z !…”

Section: Introductionmentioning

confidence: 99%

“…By using methods from algebraic dynamical systems Daniel Silver and Susan Williams, in [24], were able to show that if the first Alexander polynomial .L/ of L is nonzero then lim sup .H /!1 log jH 1 …”

Section: Introductionmentioning

confidence: 99%

“…In [1] Nicolas Bergeron and Akshay Venkatesh used this to give a new proof of the theorem of Silver and Williams in the case of a knot: the main point is that the nonnullity of the Alexander polynomial guarantees that M is`2 -acyclic, and in this case the approximation of`2 -torsion by Reidemeister torsions is known. They then show that the growth of torsion numbers is the same as that of Reidemeister torsion and obtain a generalization of the theorem for knots (see [1,Theorem 7.3]). For links with nonzero first Alexander polynomial their proof can be adapted; however the result obtained is slightly weaker than Silver and Williams'.…”

Section: Introductionmentioning

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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“…G Š Z m , for any subgroup H G we denote by X H the Galois covering of X given by 1 .H / n z X ; its Galois group is G=H . We also denote the free abelian covering of X given by ker.…”

Section: Proof Of the Main Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exponential growth of torsion in abelian coverings

Raimbault

2012

Algebr. Geom. Topol.

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“…In fact, at least with nontrivial coefficient systems, there are precise bounds on the growth of the torsion in H i (X K , Z), showing exponential growth in the case that n = 2 and F is imaginary-quadratic (while H i (X K , C) stays small), cf. [12], [50]. In other words, Conjecture I.2 predicts the existence of many more Galois representations than Conjecture I.1.…”

Section: Introductionmentioning

confidence: 96%

On torsion in the cohomology of locally symmetric varieties

Scholze

2015

Ann. Math.

134

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Abstract. The main result of this paper is the existence of Galois representations associated with the mod p (or mod p m ) cohomology of the locally symmetric spaces for GL n over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for GL n as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of Shimura varieties.Thus, we prove new results on the p-adic geometry of Shimura varieties (of Hodge type). Namely, the Shimura varieties become perfectoid when passing to the inverse limit over all levels at p, and a new period map towards the flag variety exists on them, called the Hodge-Tate period map. It is roughly analogous to the embedding of the hermitian symmetric domain (which is roughly the inverse limit over all levels of the complex points of the Shimura variety) into its compact dual. The Hodge-Tate period map has several favorable properties, the most important being that it commutes with the Hecke operators away from p (for the trivial action of these Hecke operators on the flag variety), and that automorphic vector bundles come via pullback from the flag variety.

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Torsion Invariants

Kammeyer¹

2019

Lecture Notes in Mathematics

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The asymptotic growth of torsion homology for arithmetic groups

Abstract: Abstract. When does the amount of torsion in the homology of an arithmetic group grow exponentially with the covolume? We give many examples where this is so, and conjecture precise conditions.

Cited by 133 publications

References 55 publications

Exponential growth of torsion in abelian coverings

Exponential growth of torsion in abelian coverings

On torsion in the cohomology of locally symmetric varieties

Torsion Invariants

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