Asymptotic growth patterns for class field towers

Let p be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a Z₏ -tower. These Galois groups can be considered as non-commutative analogues of ray class groups. For certain Z₏ -extensions in which a given prime above p is completely split, we prove precise asymptotic lower bounds. Our investigations are motivated by the classical results of Iwasawa, who showed that there are growth patterns for p -primary class numbers of the number fields in a Z₏ -tower.