Filtrations of profinite groups as intersections and absolute Galois groups

The general structure of absolute Galois groups of fields as profinite groups is still a mystery. Among the very few known properties of such groups are several “Intersection Theorems”, describing subgroups in standard filtrations of absolute Galois groups as the intersection of all normal open subgroups with quotient in a prescribed list of finite groups. These theorems are based on deep cohomological properties of absolute Galois groups. We will present a general “Transfer Theorem” for profinite groups, which explains what lies behind these intersection theorems.