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We study the rate of growth of normalized Hodge numbers along a tower ofabelian covers of a smooth projective variety with semismall Albanese map.These bounds are in some cases optimal. Moreover, we compute the $L^2$-Bettinumbers of irregular varieties that satisfy the weak generic Nakano vanishingtheorem e.g., varieties with semismall Albanese map). Finally, we study theconvergence of normalized plurigenera along towers of abelian covers of anyirregular variety. As an application, we extend a result of Koll\'ar concerningthe multiplicativity of higher plurigenera of a smooth projective variety ofgeneral type, to a wider class of varieties. In the Appendix, we studyirregular varieties for which the first Betti number diverges along a tower ofabelian covers induced by the Albanese variety.

L^2-Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem