Lie methods in pro-p groups.

摘要

The Lie ring method is the method of association of Lie algebras to groups. In this dissertation we use the Lie ring method to answer some questions related to pro-p groups. The crucial part of this dissertation consists of the solution of two problems; the first one related to normal zeta functions of pro-p groups, and the second one raised by Iwasawa.;The normal zeta function of a finitely generated (profinite) group G is given by the Dirichlet series zri;G (s) = n=1infinityari; n (G)n-s, where ari;n (G) denotes the number of normal subgroups of index n in G. We compute explicitly the normal zeta functions of the pro-p group SL12&parl0;F p[[t]]) and Ershov groups Q1 (s, r). As a corollary we get that Ershov groups Q1 (s, r) are normally isospectral with the group SL12&parl0;F p[[t]]), i.e. zri;SL1 2Fp t (s) = zri;Q1 s,r (s). This gives an affirmative answer to the following question: Is there an infinite family of non-commensurable normally isospectral pro-p groups?;For a positive integer n, let En denote the class of all (finitely generated) pro-p groups satisfying d(H) -- n = [G : H](d(G) -- n), for all open subgroups H of G, where d(H) denotes the minimal number of topological generators of H. In the 1980's, Iwasawa raised the question of determining all En -groups for n ≠ 1. We answer this question completely for pro-p groups of finite rank, where p > n + 1. The main result is given by the following theorem:;Theorem: Let n ≥ 2 be a positive integer and let p > n + 1 be a prime. A p-adic analytic pro-p group G belongs to the class En if and only if G is one of the following groups (up to isomorphism):;1. The abelian group G0, isomorphic to Znp , given by the presentation Gn0=x1,x 2,&ldots;,xn&vbm0; xi,xj=1 for1≤i,j≤ n. 2. The non-nilpotent (and meta-abelian) group, parametrized by s ∈ N Gn1 s=&angl0;x1,x2,&ldots; ,xn&vbm0;x1 ,xn=xps 1,&sqbl0;x2,xn&sqbr0;=xp s2,&ldots;,&sqbl0;xn-1,x n&sqbr0;=xpsn-1 and [xi, xj] = 1 for all the other combinations of i and j⟩.