Comments on Valuations Associated to Systems of Vertices/edges and the Main Theorem of Pop-stix

Let k be an arbitrary complete discrete valuation field of mixed characteristic whose residue characteristic we denote by p, k an algebraic closure of k, G k def = Gal(k/k), Σ a set of primes that contains a prime l = p, X a proper hyperbolic curve over k. for the geometrically pro-Σ quotient of thé etale fundamental group Π X of X. Thus, we have a natural surjection Π [where i ranges over the positive integers] be a cofinal system of finité etale connected Galois coverings of X with stable reduction arising from open subgroups of Π