[Paper Review] Homology torsion growth and Mahler measure

Thang T. Q. Lê|arXiv (Cornell University)|Oct 20, 2010

Geometric and Algebraic Topology15 references3 citations

TL;DR

This paper proves two conjectures in algebraic dynamics and low-dimensional topology: K. Schmidt's conjecture on the growth rate of fixed point components in algebraic dynamical systems, and a generalization of Silver and Williams' result on homology torsion growth in abelian covers of link complements. Both limits are shown to be governed by the Mahler measure of the first non-zero Alexander polynomial of the relevant module, using pseudo-isomorphism theory and tools from commutative algebra and algebraic geometry to reduce to the torsion case, with explicit sequences constructed via Bombieri–Zannier and Lawton-type results.

ABSTRACT

We prove a conjecture of K. Schmidt in algebraic dynamical system theory on the growth of the number of components of fixed point sets. We also generalize a result of Silver and Williams on the growth of homology torsions of finite abelian covering of link complements. In both cases, the growth is expressed by the Mahler measure of the first non-zero Alexander polynomial of the corresponding modules. We use the notion of pseudo-isomorphism, and also tools from commutative algebra and algebraic geometry, to reduce the conjectures to the case of torsion modules. We also describe concrete sequences which give the expected values of the limits in both cases. For this part we utilize a result of Bombieri and Zannier (conjectured before by A. Schinzel) and a result of Lawton (conjectured before by D. Boyd).

Motivation & Objective

• To prove K. Schmidt's conjecture on the asymptotic growth of the number of connected components of fixed point sets in algebraic dynamical systems over group rings.
• To generalize Silver and Williams' result on homology torsion growth in abelian covers of link complements to the case where the first non-zero Alexander polynomial is not the 0-th one.
• To establish that the growth rate of both fixed point components and homology torsion is precisely the Mahler measure of the first non-zero Alexander polynomial of the module.
• To construct explicit sequences of subgroups achieving the limiting growth rate using number-theoretic results on recurrence and density.

Proposed method

• Reduces the general case of finitely generated modules over the group ring $\mathbb{Z}[t_1^{\pm1}, \dots, t_n^{\pm1}]$ to the torsion module case via the notion of pseudo-isomorphism.
• Applies tools from commutative algebra and algebraic geometry to analyze module structure and relate it to Alexander polynomials.
• Uses the known entropy formula $h(M) = \mathbb{M}(\Delta_0(M))$ for torsion modules to establish the base case.
• Constructs explicit sequences of subgroups $\Gamma_{s,j}$ of $\mathbb{Z}^n$ with increasing minimal norm and coprime generators to approximate the limit.
• Employs the Bombieri–Zannier theorem (conjectured by Schinzel) to ensure density of certain rational directions in the unit sphere.
• Applies Lawton’s result (conjectured by Boyd) to control the growth of logarithmic Mahler measures along sequences of integer vectors.

Experimental results

Research questions

• RQ1 Does the growth rate of the number of connected components of the fixed point set in an algebraic dynamical system equal the Mahler measure of the torsion submodule’s Alexander polynomial?
• RQ2 Can the Silver–Williams result on homology torsion growth in abelian covers be extended beyond the case where the first non-zero Alexander polynomial is $\Delta_0(L)$?
• RQ3 Is the limiting growth rate of homology torsion in finite abelian covers of link complements equal to the Mahler measure of the first non-zero Alexander polynomial?
• RQ4 Can one construct explicit sequences of subgroups such that the logarithmic torsion growth rate converges to the Mahler measure of the relevant polynomial?
• RQ5 What is the role of pseudo-isomorphism in reducing general modules to torsion modules for the purpose of computing asymptotic growth rates?

Key findings

• The conjecture of K. Schmidt on fixed point component growth is proven: $\limsup_{\langle\Gamma\rangle\to\infty} \frac{\log P_\Gamma(\hat{M})}{|\mathbb{Z}^n / \Gamma|} = h(\mathfrak{tor}(M))$, with equality to the Mahler measure of the 0-th Alexander polynomial of the torsion submodule.
• The generalization of Silver and Williams' result is established: for link complements, $\limsup_{\langle\Gamma\rangle\to\infty} \frac{\log |\mathfrak{tor}_\mathbb{Z}(H_1(X_\Gamma,\mathbb{Z}))|}{|\mathbb{Z}^n / \Gamma|} = \mathbb{M}(\Delta(L))$, where $\Delta(L)$ is the first non-zero Alexander polynomial.
• For $n=1$, the $\limsup$ in both cases can be replaced by the ordinary limit, confirming convergence in the knot case.
• Explicit sequences $\Gamma_{s,j_s}$ are constructed such that the torsion growth rate converges to the Mahler measure, using integer vectors with coprime coordinates and increasing norm.
• The proof relies on the fact that $\mathbb{Z}[\mathbb{Z}^n / \Gamma_{s,j}] \otimes M$ becomes a module over a finite cyclic group ring, allowing reduction to the one-variable case.
• The convergence of the limit is shown using the density of rational directions in the sphere and the Bombieri–Zannier/Lawton theorems to control the Mahler measure along sequences.