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Let $p$ be an odd prime, $F$ be a number field and consider a uniforminfinite pro-$p$ extension $F_\infty$ of $F$ with Galois group$G=Gal(F_\infty/F)$. Let \[G=G_0\supset G_1\supset\dots \supset G_n\supsetG_{n+1}\supset \dots\] be the descending $p$ central series of $G$ and set$F_n:=F_\infty^{G_n}$. Assume that $G$ is uniform and that $F_\infty$ containsthe cyclotomic $\mathbb{Z}_p$-extension of $F$. Denote by $A_n$ the $p$-primarypart of the class group of the cyclotomic $\mathbb{Z}_p$-extension of $F_n$.The $\lambda$-invariant of $F_n$ coincides with the corank of $A_n$ as a$\mathbb{Z}_p$-module. Assume that the Iwasawa $\mu$-invariant of thecyclotomic $\mathbb{Z}_p$-extension of $F$ equal to $0$. Then, the$\mu$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $F_n$ is $0$ aswell and $A_n$ is isomorphic to$\left(\mathbb{Q}_p/\mathbb{Z}_p\right)^{\lambda_n}$. We study the asymptoticgrowth of $\lambda_n$ as $n$ goes to $\infty$.

Asymptotic growth of Iwasawa invariants in Noncommutative towers of number fields