Abelian absolute Galois groups

Generalizing a result of Wulf-Dieter Geyer in his thesis, we prove that if $K$ is a finitely generated extension of transcendence degree $r$ of a global field and $A$ is a closed abelian subgroup of $\textrm{Gal}(K)$ , then ${\mathrm{rank}}(A)\le r+1$ . Moreover, if $\mathrm{char}(K)=0$ , then ${\hat{\mathbb{Z}}}^{r+1}$ is isomorphic to a closed subgroup of $\textrm{Gal}(K)$ .