Twisted Alexander Polynomials and Representation Shifts

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the image of the commutator subgroup of the knot group.Comment: 7 pages, no figure

Mahler measure, links and homology growth

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.Comment: 13 pages, figures. Small corrections, references updated. To appear in Topolog