A Golod–Shafarevich equality and p-tower groups

TextAll current techniques for showing that a number field has an infinite p-class field tower depend on one of various forms of the Golod–Shafarevich inequality. Such techniques can also be used to restrict the types of p-groups which can occur as Galois groups of finite p-class field towers. In the case that the base field is a quadratic imaginary number field, the theory culminates in showing that a finite such group must be of one of three possible presentation types. By keeping track of the error terms arising in standard proofs of Golod–Shafarevich type inequalities, we prove a Golod–Shafarevich equality for analytic pro-p-groups. As an application, we further work of Skopin [V.A. Skopin, Certain finite groups. Modules and homology in group theory and Galois theory, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 31 (1973) 115–139 (in Russian)], showing that groups of the third of the three types mentioned above are necessarily tremendously large.VideoFor a video summary of this paper, please visit http://www.youtube.com/watch?v=13GudVNQUUI.