  For a finite abelian p-group A of rank d, we define its (logarithmic) mean
exponent to be the base-p logarithm of the d-th root of its cardinality. We
study the behavior of the mean exponent of p-class groups in towers of number
fields. By combining techniques from group theory with the Tsfasman-Valdut
generalization of the Brauer-Siegel Theorem, we construct infinite tamely
ramified towers in which the mean exponent of class groups remains bounded.
Several explicit examples with p=2 are given. We introduce an invariant M(G)
attached to a finitely generated FAb pro-p group G which measures the
asymptotic growth of the mean exponent of abelianizations of subgroups of index
n with n going to infinity. When G=Gal(L/K), M(G) measures the asymptotic
behavior of the mean exponent of class groups in L/K. We compare and contrast
the behavior of this invariant in analytic versus non-analytic groups. We
exploit the interplay of group-theoretical and number-theoretical perspectives
on this invariant and explore some open questions that arise as a result, which
may be of independent interest in group theory.
