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