Detecting pro-p-groups that are not absolute Galois groups
  We present several constraints on the absolute Galois groups G_F of fields F containing a primitive pth root of unity, using restrictions on the cohomology of index p normal subgroups from a previous paper by three of the authors. We first classify all maximal p-elementary abelian-by-order p quotients of such G_F. In the case p>2, each such quotient contains a unique closed index p elementary abelian subgroup. This seems to be the first case in which one can completely classify nontrivial quotients of absolute Galois groups by characteristic subgroups of normal subgroups. We then derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of G_F. Finally, we construct a new family of pro-p-groups which are not absolute Galois groups over any field F. 
Dave Benson, Nicole Lemire, Jan Minac, John Swallow