Estimates of the topological entropy from below for continuous self-maps on some compact manifolds

Extending our results in "Entropy conjecture for continuous maps of nilmanifolds", to appear in Israel Jour. of Math., we confirm that Entropy Conjecture holds for every continuous self-map of a compact $K(\pi,1)$ manifold with the fundamental group $\pi$ torsion free and virtually nilpotent, in particular for every continuous map of an infra-nilmanifold. In fact we prove a stronger version, a lower estimate of the topological entropy of a map by logarithm of the spectral radius of an associated "linearization matrix" with integer entries. From this, referring to known estimates of Mahler measure of polynomials, we deduce some absolute lower bounds for the entropy.