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Inspired by a remarkable work of F\'{e}lix, Halperin and Thomas on theasymptotic estimation of the ranks of rational homotopy groups, and more recentworks of Wu and the authors on local hyperbolicity, we prove two asymptoticformulae for torsion rank of homotopy groups, one using ordinary homology andone using $K$-theory. We use these to obtain explicit quantitative asymptoticlower bounds on the torsion rank of the homotopy groups for many interestingspaces after suspension, including Moore spaces, Eilenberg-MacLane spaces,complex projective spaces, complex Grassmannians, Milnor hypersurfaces andunitary groups.

Some asymptotic formulae for torsion in homotopy groups