On pro-p fundamental groups of marked arithmetic curves

Let k be a global field, p an odd prime number different from char(k) and S, T disjoint, finite sets of primes of k. Let GTS (k)(p) = G(k T S (p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and completely split at T . We prove the existence of a finite set of primes S0, which can be chosen disjoint from any given set M of Dirichlet density zero, such that the cohomology of GTS∪S0(k)(p) coincides with the étale cohomology of the associated marked arithmetic curve. In particular, cd GTS∪S0(k)(p) = 2. Furthermore, we can choose S0 in such a way that k T S∪S0 (p) realizes the maximal p-extension kp(p) of the local field kp for all p ∈ S ∪ S0, the cup-product H1(GTS∪S0(k)(p),Fp)×H 1(GTS∪S0(k)(p),Fp) → H 2(GTS∪S0(k)(p),Fp) is surjective and the decomposition groups of the primes in S establish a free product inside GTS∪S0(k)(p). This generalizes previous work of the author where similar results were shown in the case T = ∅ under the restrictive assumption p ∤ #Cl(k) and ζp / ∈ k.