Mahler measure, links and homology growth

2000/03/21 by Daniel S. Silver, Susan G. Williams, Silver, Daniel S. +1 · 2 citations
Mathematics · #37B10 (secondary) #57M25 (primary) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #math.DS #math.GT #msc:37B10 #msc:57M25

paper · pdf · doi:10.48550/arxiv.math/0003127

13 pages, figures. Small corrections, references updated. To appear in Topology

arxiv created 2001/05/25 · arxiv updated 2009/11/30

Abstract

Let l be a link of d components. For every finite-index lattice in Z^d there is an associated finite abelian cover of S³ branched over l. We show that the order of the torsion subgroup of the first homology of these covers has exponential growth rate equal to the logarithmic Mahler measure of the Alexander polynomial of l, provided this polynomial is nonzero. Our proof uses a theorem of Lind, Schmidt and Ward on the growth rate of connected components of periodic points for algebraic Z^d-actions.

Cited by

• Arithmetic invariants of torus links
• Homological torsion growth in non-normal chains of graphs of free groups

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