Homology torsion growth and Mahler measure
type: Item
instance of: scholarly article
title: Homology torsion growth and Mahler measure
zbMATH Open document ID: 1302.57005
DOI: 10.4171/CMH/332
published in: Commentarii Mathematici Helvetici
publication date: 2014-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.4199
review text: \textit{B. Kitchens} and \textit{K. Schmidt} [Ergodic Theory Dyn. Syst. 9, No. 4, 691--735 (1989; Zbl 0709.54023)] associated to any (countable) module \(M\) over the ring \(R=\mathbb{Z}[t_1^{\pm1},\dots,t_d^{\pm1}]\) an action of \(\mathbb{Z}^d\) by automorphisms of the compact (metrizable) abelian dual group of \(M\), initiating a series of results relating the algebraic and geometric properties of the module \(M\) to dynamical properties of the associated \(\mathbb{Z}^d\)-action. \textit{D. Lind} et al. [Invent. Math. 101, No. 3, 593--629 (1990; Zbl 0774.22002)] showed that if \(M\) is finitely generated then the joint topological entropy of the action associated to \(M\) is given by the logarithmic Mahler measure of \(\Delta_0(M)\), the \(0\)th Alexander polynomial of \(M\), and (under the additional hypothesis that the action is expansive) showed that the entropy is given by the growth rate of periodic points along large parallelepipeds. A fundamental feature of many algebraic \(\mathbb{Z}^d\)-actions is however that without expansiveness the set of points fixed by a lattice \(L<\mathbb{Z}^d\) will often be infinite, and Schmidt formulated a conjectural relationship between the entropy of the system associated to the torsion part of the module and the growth rate of the number of connected components of the group of \(L\)-periodic points as \(L\) goes to infinity in a suitable sense. This conjectural relationship is established here, and a related result of \textit{D. S. Silver} and \textit{S. G. Williams} [Topology 41, No. 5, 979--991 (2002; Zbl 1024.57007)] on the growth of homology torsion of finite abelian coverings of link complements is generalized.
reviewed by: Thomas B. Ward
Mathematics Subject Classification ID: 57M10
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 37B50
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 6361418
zbMATH Keywords: Mahler measure
zbMATH Keywords: Alexander polynomial
zbMATH Keywords: algebraic dynamical system
zbMATH Keywords: pseudo-isomorphism
zbMATH Keywords: periodic point
author: Thang T. Q. Le
MaRDI profile type: Publication
arXiv ID: 1010.4199
OpenAlex ID: W2963869043
description: scientific article; zbMATH DE number 6361418