Cohomological dimension of some Galois groups

Suppose that $l$ is a prime number, $k$ is an algebraic number field containing a primitive root $\zeta_l$ ($\zeta_4$ if $l=2$), $S$ is a finite set of places of $k$ which contains all divisors of $l$, $K$ is the maximal $l$-extension of $k$ unramified outside $S$, $k_\infty$ is an arbitrary $\Gamma$-extension of $k$, and $H=G(K/k_\infty$. In this paper we find necessary and sufficient conditions for the group $H$ to be a free pro-$l$-group. We also obtain a description of all $\Gamma$-extensions $k_\infty/k$ having the property that any place of $k$ has a finite number of extensions to $k_\infty$. We prove that, in some sense, such $\Gamma$-extensions make up the overwhelming majority of all $\Gamma$-extensions.
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