On tamely ramified infinite Galois extensions

For a number field $K$, we consider $K^{\mathrm{ta}}$ the maximal tamely ramified algebraic extension of $K$, and its Galois group $G_K^{\mathrm{ta}}=\mathrm{Gal}(K^{\mathrm{ta}}/K)$. Choose an odd prime $p$. Our guiding aim is to characterize the finitely generated pro-$p$ quotients of $G_K^{\mathrm{ta}}$. We give a unified point of view by introducing the notion of stably inertially generated pro-$p$ groups $G$, for which linear groups are archetypes. This key notion is compatible with local tame liftings as used in the Scholz–Reichardt theorem. We realize every finitely generated pro-$p$ group $G$ which is stably inertially generated as a quotient of $G_K^{\mathrm{ta}}$. Further examples of groups that we realize as quotients of $G_K^{\mathrm{ta}}$ include congruence subgroups of special linear groups over $Z_p⟦T_1,\text{…},T_n⟧$. Finally, we give classes of groups which cannot be realized as quotients of $G_Q^{\mathrm{ta}}$.