Mahler measure, links and homology growth

Abstract

Let l be an oriented link of d components with nonzero Alexander polynomial Δ( u 1,…, u d ). Let Λ be a finite-index subgroup of H 1(S 3−l)≅ Z d , and let M Λ be the corresponding abelian cover of S 3 branched along l. The growth rate of the order of the torsion subgroup of H 1( M Λ ), as a suitable measure of Λ approaches infinity, is equal to the Mahler measure of Δ.