Normal subgroups of free constructions of profinite groups and the congruence kernel in the case of positive characteristic

We prove the analogue of the Kurosh subgroup theorem for closed normal subgroups of free constructions of profinite groups and also corresponding analogues of abstract structural results for closed normal subgroups of more general free constructions of profinite groups (amalgamated free products, HNN-extensions). The structure theorem is used to obtain a description of the congruence-kernel $C$ of the arithmetic lattice $\Gamma$ of the group of $K$-rational points $G=\mathbf G(K)$ of a semisimple connected algebraic group $\mathbf G$ of $K$-rank 1 over a non-Archimedean local field $K$.