Exponential growth of torsion in abelian coverings

Abstract

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

The asymptotic growth of torsion homology for arithmetic groups

“…Applied to cyclic covers of a knot complement, Theorem 7.3 translates into the theorem of Silver and Williams [65] mentioned in the introduction. It is possible to prove a version of (7.3.2) for m > 1 replacing lim by lim sup but it is a bit more subtle (see [43,59]); also, one can establish (7.3.2) for general m if we suppose that each det ∆ j is everywhere nonvanishing on (S 1 )…”

Some non-abelian covers of knots with non-trivial Alexander polynomial

“…Raimbault [27], where the towers of cyclic covers are not exhaustive, but a similar behavior is exhibited. We will conclude the paper providing tables of computations and highlighting certain relationships between the torsion subgroups of H 1 (X α ; Z) and H 1 (X n ; Z).…”

Growth of regulators in finite abelian coverings

“…Theorem 2 Suppose C is a finitely generated based free ZOE -complex with D Z n and H The question (and some form of the conjecture) about the growth rate of regulators was first raised in [1], where, among other things, a special case of Theorem 2 was established: It was proved that if D Z n and runs the set of sublattices of the form kZ n , then (3) holds. The proof there can be modified to include the case when runs the set of uniform sublattices, as defined in [16]. Our result removes any restriction on .…”

“…For a detailed discussion of (6) and its generalizations to lattices in ZOEZ n , see Raimbault [16].…”