Generators of Simple Lie Algebras and the Lower Rank of Some Pro-P Groups

Abstract

Let ℊ be a simple classical Lie algebra over a field F of characteristic p > 7. We show that > d (ℊ) = 2, where d(ℊ) is the number of generators of ℊ. Let G be a profinite group. We say that G has lower rank ≤ l, if there are {G α} open subgroups which from a base for the topology at the identity and each G α is generated (topologically) by no more than l elements. There is a standard way to associate a Lie algebra L(G) to a finitely generated (filtered) pro-p group G. Suppose L(G) ≅ ℊ ⊗ tF p [t], where ℊ is a simple Lie algebra over F p , the field of p elements. We show that the lower rank of G is ≤ d (ℊ) + 1. We also show that if ℊ is simple classical of rank r and p > 7 or p 2r 2 − r, then the lower rank is actually 2.