On the Growth of Subalgebras in Lie p-Algebras

Let L be a finitely generated Lie p-algebra over a finite field F. Then the number, an(L), of p-subalgebras of finite codimension n in L is finite. We say that L has PSG (polynomial p-subalgebras growth) if the growth of an(L) is bounded above by some polynomial in F n. We show that if L has PSG then the lower central series of L stabilises after a finite number of steps. On the other hand, if L is nilpotent then L has PSG. We deduce the following group-theoretic result. Let G be a group and let G denote a pro-p completion of G. Then the associated Lie p-algebra p(G) of G has PSG if and only if G is a p-adic analytic Lie group.