Mahler measure, links and homology growth

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 S^3 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.