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We study the growth of the order of torsion subgroups of the homology in a tower of finite abelian coverings. In particular, we prove that it is exponential for when the tower converges to the maximal free abelian cover of a link complement when the first nonzero Alexander polynomial has positive logarithmic Mahler measure.doi:10.2140/agt.2012.12.1331 fatcat:nkvolvgsgrehjlbbel7bbwgjou